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Generalized Boolean Algebras and Related Problems. Representation Theorems
C H A P T E R VI
Generalized B o o l e a n Algebras and R e l a t e d Problems. R e p r e s e n t a t i o n T h e o r e m s From the late 19th century the tendencies enabling a scholar to view different mathematical theories from the unified standpoint began to gain prevalence and reached rather a high level of development. This observation is confirmed by many studies, of which, keeping in mind our interest, we would like to mention M. H. Stone's remarkable theory of representations that brought together the plain algebraic structures of Boolean algebras with topological type structures, that is, with a special class of TS's [246], [247]. For quite a long time the proofs of theorems had been based on the Boolean algebra laws. G. Boole was the first to formalize these laws [36], [37], thereby providing a powerful incentive to the development of symbolic logic [28]. Subsequently, Boolean algebras became widely adopted in such areas as probability theory, functional analysis, topology, Stone's representation theory, and others. Chapter VI continues and develops the study begun in [91], [95], [100]. If, presumably, the definition of a Boolean algebra employs the notion of a partially ordered set (briefly, poset), whereas a general definition of a TS is based on the notion of a metric, then by considering a more general situation with quasi metrics one may obtain the notion of a BS. A further development of the theory of BS's made it possible to introduce and study an algebra of a new type that is based on a nonordinary variant of a quasi ordered set and the corresponding representation of which brings one to BS's. We believe that this algebra, which we call a generalized Boolean algebra, is interesting not only as the subject of independent research, but also can be used as an important tool in establishing its various relationships with other areas of mathematics. Moreover, since a quasi ordered set of special type, which is the union of two posets forms the basis for defining both generalized Boolean algebras and double Boolean algebras [134], [261] and the latter algebras have already been used in metamathematical studies, due to this common basis, generalized Boolean algebras can also be used in mathematical logic. In Sections 6.1-6.6 we introduce and study generalized Boolean algebras and some other related important questions. To this end, we first define a generalized ordered set (briefly, goset) in terms of a quasi ordered set which is a union of two posets whose partial orders are induced by the quasi order relation and whose intersection contains only zero and unit elements. Furthermore, to define a generalized lattice, for any subset of a goset we introduce the important notions of i~.-inf and i(;-sup. A different approach to defining a generalized lattice as 193
194
VI. Generalized Boolean Algebra and Related Problems
an algebra, satisfying in particular, the generalized associativity and generalized absorption laws, is also used. Theorem 6.1.15 shows how we can pass from the generalized lattice, defined by means of a goset, to the generalized lattice using an algebra and vice versa. The results obtained make it possible to connect the posers, making up the goset, by the one-to-one correspondence and thus to introduce the basic notion of a generalized Boolean algebra. The one-to-one correspondence between the posers is determined by the generalized identity operator which plays an important role in our further studies because, on the one hand, it determines the interdependence of two structures forming a generalized Boolean algebra and, on the other hand, allows us to characterize a generalized Boolean algebra in a different way. We also discuss some other operators and binary operations and give a few interesting examples of generalized Boolean algebras. Furthermore, we assign the V-formation and the corresponding strong amalgamation to every generalized Boolean algebra, and we define a generalized quasi measure on a generalized Boolean algebra and a bitopological generalized Boolean algebra. In Sections 6.2 and 6.3, we obtain key generalizations of an ideal and its variety such as a prime, that is, a maximal generalized ideal, left and right principal generalized ideals, and generalized ideals generated by different pairs of families of sets. We introduce and investigate the notion of a generalized Stone family of prime generalized ideals as well as the notion of a generalized dual to a generalized ideal, that is, the notion of a generalized filter together with its variety. The relation between generalized components and generalized ideals is also established. Other results obtained in the first half of this chapter concern the important notions of generalized homomorphic and generalized isomorphic maps, (i,j)-atoms, and p-atomic generalized Boolean algebras and generalized Boolean factor algebras. The remaining three sections deal with the bitopological modification of Stone's representation theory, generalized complete generalized Boolean algebras, and generalized Boolean rings. We introduce the notions of a generalized field of sets, its reduced version, and generalized field representation. Based on the results of Section 6.2, it is proved, in particular, that there exists a one-to-one correspondence between the reduced generalized field representations of a generalized Boolean algebra .4 and the generalized Stone families of prime generalized ideals of ,4. Thus every generalized Boolean algebra becomes generalized isomorphic to the reduced generalized field of sets. With every generalized field representation of a generalized Boolean algebra we associate a BS and give the necessary and sufficient conditions under which this BS becomes FHP-compact. Our main result is the generalized version of Stone's representation theorem which states that under special hypotheses there exists a one-to-one correspondence between generalized Boolean algebras and p-zero dimensional, p-Hausdorff and FHP-compact (also called Boolean) BS's. We would like to recall that an attempt at connecting the bitopological structure with a generalized algebraic structure was for the first time made in [91]. Throughout this chapter, the term "generalized" will be denoted for brevity by the symbol "G" and k, 1 c { 1, 2} , k ~r 1.
6.1.
Gosets,
Generalized
Lattices,
...
195
6.1. Gosets, Generalized Lattices, Generalized Boolean Algebras, and the Corresponding Operations Definition 6.1.1. A quasi ordered set (P, 4 ) is said to be a goset if P P1 U P2, where (P~, <) are nonempty posets and the restrictions 4 ]P~ -<_. i
i
It is clear that every poset is a goset, but not conversely.
Example 6.1.2.
(X, T1,T2) be
Let
a nonempty BS.
Then the families
j-OZ)(X) r ;g # (i, j)-OT)(X) are partially ordered by the set-theoretic operation inclusion and the binary relations, defined on the sets
P - j-oz)(x)
o (i, j ) - o z ) ( x )
as follows A1, A2 c P, A1 4 A2 <--> Aa c_ ~-j clA2 are the relations of quasi order on P. Moreover,
(P - j - o z ) ( x )
o
j)-oz)(x), 4 )
are gosets. Indeed, by (2) of Proposition 1.3.10, the restrictions 4 A1 4 A2 ~
I(i,j)-oz)(x) give
A1 C_ Tj clA2 <---> Tj clA1 C_ Tj clA2 <----> A1 C_ A2
(i,j)-oz)(x) =c_=<. Similarly, by the topological correspondence of (2)
and so ~
of Proposition 1.3.10, 4
i
Ij-oz)(x) =c_=<_. J
The quasi order 4 , defined in Example 6.1.2, shows that (P (1, 2)-OD(X), 4 ) in Example 1.3.5 is a goset. For the sets
-
2-or(x)u
{a, b} c 2-OT)(X) and {at c (1, 2)-OT)(X) we have {a,b} 4 a and {a} 4 {a,b}, but {a,b}-r {a}. Thus t h e g o s e t (P, 4 ) is not, generally speaking, a poset. If (P, 4 ) is a goset, then for any pair x, y c P the notation x -~ y means z 4 Y and z -r y. Therefore for z c P1 \ P2, Y c P2 \ P1, we have z -< y <---> x 4 Y, whereas for z, y c Pi, we have z -~ y <---> z < y. i
Therefore in Example 1.3.5, we have {a, b} -~ {a} and {a} -~ {a, b}. e b
f
.. -TPx, --- -'- -'-
a
// J
d
f
1 1
Diagram 1
In
Diagram
goset (P = {a,b,c,d} and Pe - {c,e,f}, the small circles denote the elements; the circles, corresponding to the elements x, y c Pi, are connected by the solid line while for x c Pi and y c PO, the connection is broken.
{a,b,c,d,e,f},4
1
of
the
), where P1
196
VI. Generalized Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
The quasi order r e l a t i o n x 4 y for x E Pi, y E Pj means that y is over x or x and y lies on the same horizontal line; whereas the restriction 4 IP, - < gives that x < y if y is over x or x - y. i
It clearly follows that for a E P1, f E P2, we have a -< f and f -< a, a ~ f. Note that in the sequel, the indices i~; and Ja satisfy as usual the conditions i a , j a C {1,2}, i~ =fi JG, and, moreover, accentuate a more general character of the corresponding notions as compared with usual structural ones. D e f i n i t i o n 6.1.3. Let (P, 4 ) be a goset and A c_ P be a subset such that A N P1 - A1 ~ ~ ~ A2 - A N P2. Then A is called to be a G.chain (G.convex set) if a 4 b o r b 4 a for each pair a, b E A ( a , b c A a n d a 4 c4 bimplycEA). If A c_ P is a G.chain (G.convex set), then A~ are chains (convex sets) in the Usual sense.
An element xE Pi is an ia-lower (/,-upper) bound of a subset A - A 1 U A 2 c_ P, where ( P - P 1 U P 2 , 4 ) isagoset, ifx <_aforeachac Ai a n d x 4 b f o r e a c h i
b E A j (a <_ x for each a c Ai and b 4 x f o r e a c h b c A j ) . i The sets of all ia-lower (/,-upper) bounds of A are denoted by i a - l ( A ) - { x C Pi " x < a, a c Ai and x 4 b, b c A j } i
(ia-u(A)
- { x E Pi " a <_ x, a c Ai and b 4 x, b E A j } ) . i
Moreover, l~;(A) - I ~ - / ( A ) U 2 , - l ( A ) - { x E P " a ~ x, a E A } (ua(A)
- l a - u ( A ) U 2 a - u ( A ) - {x C P " x 4 a, a C A } ) .
Furthermore, if (P, ~ ) is a goset, then a subset A c_ P is G.normal (or A is G.normally included in P) if a c A, x r P and x 4 a imply x r A. Hence, if A is G.normal, then l~; (A) c A. D e f i n i t i o n 6.1.4. An element x c Pi is called an /a-smallest (/a-largest) element of the goset (P - P1 U P2, 4 ) if x is the smallest (largest) element of the poset (Pi, <) such that x 4 a (a 4 x) for each a c Pj. i
The i~;-smallest element (if it exists) will be denoted by the symbol ic-@ (i~-zero element), while the io-largest element (if it exists) will be denoted by the symbol i~jm (iG-unit element). Since the existence of smallest (largest) elements of posets (P~, <) is the neci
essary condition for the existence of i a-smallest (i~-largest) elements of a goset ( P - P1 UP2, 4 ), the uniqueness of/a-smallest (i(~-largest) elements (if they exist) is clear. In the context of the arguments presented above let us consider the following elementary, but necessary, example.
6.1. G o s e t s , G e n e r a l i z e d
Lattices,
...
197
E x a m p l e 6.1.5. Let the set P - { - 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4 } be linearly ordered (by the usual order) and, therefore, quasi ordered as the set-theoretic union of linearly ordered and thus partially ordered sets P1 - { - 4 , - 3 , - 2 , - 1 } , P2 - {1, 2, 3, 4}. Clearly, ( P - P1 U P2, 4 ), where x 4 y ~ x <_ y for each pair x, 9 c P and for the usual order <, is a goset. The element 1 c P2 is the zero element of P2, but it is not the 2a-zero element of P, while - 4 c P1 is the l a - z e r o element of P. Similarly, 4 c P2 is the 2a-unit element of P, while - 1 c P1 is the unit element of P1, but it is not the 1a-unit element of P. In the sequel we shall consider, in general, gosets ( P - P1 U P2, 4 ) such that (P1, <_) and (P2, <) are posets with common zero and unit elements denoted by G 1
2
and e, respectively; obviously, then O - l a - O - 2a-(~ and e - l a - e - 2a-e. Now we introduce the notion which is highly important for our later considerations. D e f i n i t i o n 6.1.6. Let (P, 4 ) be a goset and A c_ P be a subset such that A1 r 2~ r A2 - A N P2. Then an i(;-infimum of A (briefly, i~;-inf A) (an i , - s u p r e m u m of A (briefly, i , - s u p A ) ) is an element from i(;-l(d) ( i , - u ( A ) ) , satisfying the conditions below: A n P1 -
(1) i(;-inf A < _ inf A~ (sup A~ < / , - s u p A) and i
i
i(~-inf A 4 inf Aj
( sup Aj 4 i , - s u p A ) .
(2) I f x c P~, x <_infA~ (supA~ <_x) a n d x 4 a (a 4 x) for e a c h a c Aj, i
then x _< i , - i n f A (iG-sup A < x). i
i
(3) I f x c Pj, x 4 a (a 4 x) for e a c h a c A i then x 4 i(;-inf A ( / , - s u p A 4 x).
andx
(supAj < x), 5
It is obvious that the existence of inf A1 and inf A2 (sup A1 and sup A2) is the necessary condition for the existence of i , - i n f A ( / , - s u p A). Moreover, the notions of i~;-inf A (i~-sup A) will be meaningful if and only if their uniqueness is proved when they exist. As an example, let us consider the case of i~-sup A. If we assume that x and y are both i~;-sup A, a - sup Ai c Pi and b - sup Aj c Pj, then by (1) of Definition 6.1.6, we have a <_ x, b 4 x and a < y, b 4 y. It is obvious that z 4 x for i
each z E Aj since z <_ b ~
i
z 4 b. Therefore x E Pi, a < x and z 4 x for each
2
i
z c Aj. But y - / ( ; - s u p A and from (2) of Definition 6.1.6 it follows t h a t y <_ x.
i On the strength of the same argument, we conclude t h a t x _< y and thus x - y. i
R e m a r k 6.1.7. Let (P, 4 ) be a goset and A c_ P be a subset such that A n P1 - A1 7s 2~ r A2 - A n P2. First, we shall show t h a t j~;-inf A 4 i(;-inf A and j(~-supA 4 /(;-supA. Our consideration involves only the case of infimum since the case of supremum can be proved similarly. By (1) of Definition 6.1.6, jc;-inf A 4 inf Ai. But inf Ai < x ,e---->, inf A~ 4 x for each x E A~. Therefore i
198
VI. Generalized Boolean A l g e b r a and R e l a t e d P r o b l e m s
jc-inf A 4 x for each x E Ai. Further, by (1) of Definition 6.1.6, jc-inf A < inf Aj.
7
Since jG-inf A c Pj, jc;-inf A 4 x for each x c A~ and jG-inf A < inf Aj, by (3) of
7
Definition 6.1.6, we have j~-inf A 4 ia-inf A.
Moreover, for A=A~ c_ Pi, Definition 6.1.6 reads as follows: i~-inf A (i~-sup A) is an element inf A E Pi (sup A c P~) such that x E Pj and x 4 a (a 4 x) for each a E A imply x 4 inf A (sup A 4 x). Without loss of generality, let us consider the case o f / , - s u p A. Since A - Ai, we have Aj c_ Ai. By (1) of Definition 6.1.6, sup A _< i~-sup A. On the other hand, assuming that x c Pi and sup A _< x, we obtain
i
i
supAj <_ s u p A since Aj c_ Ai - A c_ Pi. i Besides, sup A < x implies i sup Aj <_ x and so a <_ x for each a E Aj, i
that is a 4 x for each a E Aj. Therefore xCPi,
s u p A _ < x implies a 4 i
x for each a c A j
and, by (2) of Definition 6.1.6, we have ia-sup A _< x. Hence sup A <_ x implies i~;-sup A _< x for each x E Pi i
i
and, therefore, i(~-sup A < sup A i
so that i ( ; - s u p A - s u p A.
Finally, assuming that x c Pj and a 4 x for each a c A, we obtain a 4
x ~
a < x
for each
a
E
Aj
J
since Aj c_ A and hence sup Aj _< x. Thus J xcPj, a 4 x for each a c A
imply s u p A j _ < x J
(since Aj c_ A - Ai) and, by (3) of Definition 6.1.6, i~;-supA 4 z. But, as we have seen above, i a-sup A - sup A and, therefore, x c Pj, a 4 x for each a c A imply sup A 4 x. Clearly for A c_ Pi, the existence of inf A (sup A) is the necessary condition for iG-inf A (it-sup A) to exist and, therefore, iG-inf A (it-sup) (if the latters exist) are unique by virtue of the uniqueness of inf A (sup A). Nevertheless we want to show by giving a simple example below that the existence of inf A (sup A) is not sufficient for i~j-inf A (iG-sup A) to exist. E x a m p l e 6.1.8. Let X - {a, b, c, d, e}, ~-1- {2~, {a}, {b}, {c}, {d}, {a, b}, {a, c},
{a, d}, {b, c},{b, d},{c, d}, {a, b, c},{a, b, d},{a, c, d},{b, c, dI,{a, b, c, d},{a, b, c, e}, X}, and 7-2 be the discrete topology on X. By Example 6.1.2, ( 1 - O r ( X ) U (2, 1)-OZ)(X), 4 ) is a goset.
6.1. Gosets, Generalized Lattices, ...
199
Let us consider the family of sets ~41 = {{a, d}, {b, d}} C 1-Or(X) and the set {d, e} c (2, 1)-OZ)(X). Then {d, e} = {a, d, e} n {b, d, e} = 7-1 cl{a, d} n 7-1 cl{b, d} so that {d,e} ~ {a,d}, {d,e} ~ {b,d}. But inf .At -- {a, d} n {b, d} = {d} E co 7-1. Therefore {d, e} 4 inf A1 does not hold, that is, 1a-inf.41 does not exist. Of considerable importance is E x a m p l e 6.1.9. Let (X, 7-1,7-2) be a nonempty BS. As is well known, 7-i with the set-theoretic operation inclusion as partial orders are complete distributive lattices with respect to the lattice operations
AUs-7-iintNUs and V Us- Uus sES
sES
sES
sES
for every subfamily N - {Us}s~S c_ 7-i. One can easily satisfy oneself that the maps T~ int 7-j c l ' T i ~ ~-i have the following properties: (1) U C_ 7-i int 7-j cl U for each U c Ti. (2) If U c_ V, then T~ int 7-j cl U c_ 7-i int Tj cl V for each pair U, V c 7-i. (3) 7-~int 7-9 el 7-i int 7-r cl U - 7-i int 7-j el U for each U c 7-i. Thus 7-i int 7-j cl are closure operators on 7-i in the sense of [105, p. 13]. Moreover, the "closed" (in the sense of such operators) elements of 7-i are the (i, j)-open domains of (X, 7-1,7-2), and by Theorem 8.1 from [105], the families (i,j)-(.gz)(X) form complete lattices with the same partial order as that of 7-i, while by Theorems 8.1 and 8.2 from [105] the lattice operations in (i,j)-O~?(X) are written aS
A' Us --7-iint
~') Us
sCS
sES
and V' Us --7-,int7-jcl U Us sES
sES
for every subfamily b / - {Us}scs C (i,j)-O2P(X). Note that the latter equality V-s sES
d. U sES
is actually the essence of Lemma 6.1.2, proved in [2381. As Example 6.1.8 shows, for a BS (X, 7-1 < 7-2) and the goset (1-OZ)(X)U (2, 1)-OZ)(X), 4 ), there does not always exist l c - i n f A, where A c 1-O~P(X) is some subfamily. Furthermore, in that case for the BS (X, 7-1 < 7-2) neither does there always exists l c - i n f A for a subfamily .4 c 1-O79(X)U (2, 1)-OZ)(X) such that A N 1-OZ)(X) - J[1 ~ ~ r ~ 2 - - W ~ A (2~ 1)-OZ)(X). Indeed, if we assume that l a - i n f A exists for any such A and take the BS from Example 6.1.8 as a BS (X, 7-1 < 7"2) and the family {{a, d}, {b, d}} U (2, 1)-O:D(X) as A, then by virtue of the second part of Remark 6.1.7, there must exist l c - i n f A1, where A1 - {{a, d}, {b, d}} c 1-OD(X), which contradicts Example 6.1.8. Therefore ic-inf A, where .4
n 1-OT)(X)
- A1 -~ 2~ r A2 - A
n (2, 1)-OT)(X),
200
VI. Generalized Boolean Algebra and Related Problems
does not always exist for a BS (X, 7"1 < 7"2). However, we shall show below that for a BS (X, 7"1 < 7"2) and the goset (1-O2)(X) U (2, 1)-OD(X), 4 ) there always exist 1a-sup A, 2c-inf A and 2a-sup A (which are unique), where either A c_ 1-OD(X) or A _c (2, 1)-OD(X) although 2c-sup.4 exists even for .4 c 1 - O D ( X ) U (2, 1)-O~D(X) and A
N 1-O2)(X) - A1 7s 2~ 7s .42 -- A N (2, 1)-OT)(X).
Choose any subfamily ,4 c_ 1-OT~(X). Again applying Theorems 8.1 and 8.2 from [105], we find that there always exists sup.4 - V 1 A -- 7"1 int 7"1 el U A. If AEA
AEA
there exists a set B E (2, 1)-(.9Z)(X) such that A 4 B, that is, A c_ 7"1c l B for each A c .4, then 7"1 int 7"1cl U A c_ 7"1cl B so that sup A 4 B. Therefore sup A ACA
is l a - s u p A. Next, we shall consider a subfamily .4 c (2, 1)-OZ)(X) and show that
A2A-7"2int N A .~d
infA--
ACA
su A- V2A-7"2intT"lCl U
AC.A
AC.A
A
AEA
always exist. It is assumed that B c 1-OD(X) is a set such that B 4 A, that is, B C 7"1 el A for each A E A. We are to prove that B 4 inf.4 so that B c_ 7"1 cl 7"2 int n A. If we assume the contrary, that is, AEA
B n
(X \7-1C17"2 int N
.,
(X \7"2int N
then B N
AEA
AEA
But B 91-O~P(X), A E (2, 1)-OZ)(X) and, therefore, for each A c A the inclusion g C 7"1 clA implies 7"1 int 7"1 cl g C 7"2 int 7"1 cl A, that is, B c_ A for each A c A since T 1 C 7-2. Hence BC_ n
A and thus B C_ 7-lint n
AEA
Ac-7-2int n
ACA
which contradicts the inequality B O (X \ 7-2 int n 2a-inf A. Now we suppose that B c 1- 9 A E A. Then the obvious inclusion
A,
AEA
A) r z . Therefore inf A is
AEA
and A 4 B, that is, A c_ 7-1 cl B for each
s u p A - 7"2int T1 cl U A c_ 7-1 cl B AEA
implies that sup A is 2c-sup A. Finally, for a BS (X, T1 < 7"2) let us consider a subfamily A c_ 1-OD(X) u (2, 1)-O:D(X) such that .4 5 1-O2)(X) - A1 7~ ~ 7~ .42 - A n (2, 1)-OZ)(X). We shall show that the set 72 int T 1 cl (72 int T 1 clsup A1 U s u p A2) is 2a-sup A. For this it is sufficient to check that (1)-(3) of Definition 6.1.6 are fulfilled. (1) The inclusions sup A2 C_ 2a-sup A and sup A1 c_ T1 C12a-sup A are obvious.
9
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202
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
Theorem 6.1.11. A goset (L, 4 ) is a G.lattice if and only if for any nonempty finite subset A c L, there exist i(:-inf A and/(;-sup A.
Proof. Clearly, it suffices to prove only the necessary part of the theorem. First we assume t h a t A c Li. If A = {x}, then the equalities i(j-inf A = i~.-sup A = x are obvious. Now let A = {x, y, z}. The proof will be carried out for infimum and s u p r e m u m simultaneously. By (1) of Definition 6.1.10, we have v = ic-inf(x,y ), t = i , - s u p ( x , y) and, therefore, there exist w = i , - i n f ( v , z), r = / ( ; - s u p ( t , z). Let us show t h a t w = i~;-inf A and r = / , - s u p A. We easily find t h a t w = inf A and r = sup A in the usual sense. Assume t h a t a c Lj is an element such t h a t a~
x, a ~
y and a ~
z
(x~
a, y ~
a and z ~
y
a, y 4
and v - i ( ~ - i n f ( x , y )
a).
Since a4
x, a 4
(x4
a)
(t-/(;-sup(x,y)),
by a) of (1) of Definition 6.1.10, a 4 v (t 4 a). By virtue of the same a r g u m e n t a4
v, a 4
z (t4
a, z 4
a)
imply a 4
w (r4
a).
This completes the proof for A - {x, y, z}. If A - { x 0 , x l , . . . , x n - 1 }, where n > 1, then the element i(:-inf((--, i(:-inf(i,-inf(xo, X l ),
X2),...,
Xn-1 )
( / ( : - s u p ( - . - i c : - s u p ( i , - s u p ( x o , X l ), X2 ), . . . , X n - 1 ) ) is i , - i n f A (/(:-sup A), which is proved by induction. Now we suppose t h a t
A N L ~ - A1 r 2~ r A2 - A N L 2 . Clearly, Ai is finite and, as in the first case, there exist x - j , - i n f Aj
(x - j(:-sup Aj ) and y - i~:-inf Ai ( y - / ( : - s u p Ai ).
Since (L, 4 ) is a G.lattice, by (2) of Definition 6.1.10, there exist v - i(j-inf(x, y) and t - i , - s u p ( x , y). Let us prove t h a t v - i , - i n f A and t - / , - s u p A in the sense of Definition 6.1.6. Indeed, by a) of (2) of Definition 6.1.10, v <_ y (y _< t), v 4 x i
i
(x 4 t) so t h a t v <_ inf Ai - i~;-inf Ai ( i(:-sup Ai - sup Ai _< t ) i
i
and v 4 inf Aj - j~;-inf Aj
(j(;-sup Aj - sup Aj 4 t).
Further, let
zcLi,
z4x
( x 4 z), z < y i
(y
T h e n by b) of (2) of Definition 6.1.10, z < v (t < z). Finally, if i
zcLj,
z
(x
i
and z 4
y ( y 4 z),
then by c) of (2) of Definition 6.1.10, z ~ v (t ~ z). Thus the elements v c Li and t E Lj satisfy all the conditions of Definition 6.1.6 so t h a t v - i , - i n f A (t - / ( ; - s u p A). D
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204
VI. Generalized Boolean Algebra and Related Problems
Then
(Xl Vj x2 V j . . . V j Xn) Vi (Yl ViY2 V i " " ViYn) -((x 1 Vj x 2 Vj " " Vj Xn_l) Vj xn) V i ((Yl Vi Y2 V i " " Vi Yn-1) Vi Yn) --
(x n Vj (Xl Vj x2 V j " ' V j
-
-
-
Xn_l) ) V i ((Yl Vi Y2 V i " " Vi Yn-1) Vi Yn) (Xn Vi Yn) Vi ((Xl Vj x2 V j " ' V j Xn_l) Vi (Yl Vi Y2 V i " " Vi Yn-1)) --
-
-
-
-
(Xn Vi Yn) Vi ((Xl Vi Yl) Vi (x2 Vi Y2) V i " " Vi (X,n-1 Vi Y'n-1)) -= (Xl Vi Yl) Vi (x2 Vi Y2) V i " " Vi (xn V~ Yn).
D e f i n i t i o n 6.1.13. Let s = {L = L1 U L2, A1, Vl, A2, V2} be a G.lattice and L~ c L~. Then s "- {L' - Ltl U L~, A1, V1, A2, V2} is said to be a G.sublattice of Z; if a c L~ U L~, b c L~ implies that a A~ b, a V~ b c L~. As we shall see in Sections 6.2 and 6.3, this notion is closely associated with the notions of G.ideal and G.filter. R e m a r k 6.1.14. Note t h a t by Definition 6.1.12, if an element x E Li is on the right-hand side of the operations Ai and Vi, then the definition of the domains of A~ and Vi necessarily implies that x c Li. This fact accounts for the absence of laws, corresponding to the commutativity laws in the well-known sense, t h a t is, the absence of the laws for Ai and Vi, corresponding to L2 when (x, y) c Lj x Li, though as we shall see in the sequel, such G.commutativity laws are available for G.lattices with the G.zero and G.unit elements. Our next important theorem shows the ways how we can pass from the G.lattice, defined by a goset to the G.lattice, defined by an algebra and vice versa. According to this theorem, no m a t t e r in which order we perform such pass-overs initially, they do not change the initial objects. Note that to prove the theorem, we do not need to assume that the G.lattice have the G.zero and G.unit elements. Theorem
6.1.15. The following conditions hold:
(1) Let a goset s = (L, 4 ) be a G.lattice. A s s u m e that x A~ y = i , - i n f ( x , y) and x Vi y = i~;-sup(x, y) for each pair (x, y) e (ni x ni) U (nj x Li). Then the algebra/~a = {L, A1, Vl, A2, V2} is a G.lattice. (2) Let an algebra s = {L, A1, V1, A2, V2} be a G.lattice. A s s u m e that y ~ x if and only if x Ai y = y for each pair (x, y) c (Li x Li) U (Lj x Li). Then 12q = (L, ~ ) is a goset which is a G.lattice. (3) If a goset s = (L, ~ ) is a G.lattice, then (s = s (4) If an algebra 12 = {L, AI~ Vx~ A2~ V2} i8 a G.lattice, then (s __ s
Proof. Conditions (1) and (2) will be proved first for i~-inf, that is, for Ai, and then f o r / , - s u p and so for Vi. Such a proof covers all the cases and we avoid going into a long tiresome discussion. (1) Let a goset Z; = (L, 4 ) be a G.lattice. By Definition 6.1.10 the binary operations Ai, Vi: (Li x L~)U (Lj x L~) ~ Li are defined as x Ai y = i , - i n f ( x , y) and x V~ y = ic~-sup(x , y). Let us show that the algebra s = {L, A1, V1, A2, V2}
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6.1. G o s e t s , G e n e r a l i z e d L a t t i c e s , . . .
207
Therefore, by b) of (2) of Definition 6.1.10, i , - s u p ( y , z) < i , - s u p ( j , - s u p ( x ,
y), z)
i
and thus i ( ; - s u p ( x , i ( ; - s u p ( y , z) ) <_ i G - s u p ( j ( ; - s u p ( x , y), z) i
so t h a t i.-sup(j.-sup(x,
y), z) - / , - s u p ( x , / . - s u p ( y ,
z)).
But this equality means t h a t (x Vj y) V~ z - x V~ (y V~ z). In a similar manner, using Definition 6.1.10, we can prove t h a t (xAjy) A~z--xA~(yAiz)
for x, y C L j ,
z C L~,
or x, z c L~,
y c Lj.
T h u s condition GL3 of Definition 6.1.12 is satisfied. If x c Lj, y c L~, t h e n by a) of (2) of Definition 6.1.10, i(;-inf(x,y) 4 x and, therefore, b) of (2) of the same definition implies j ( x - s u p ( i ( ~ - i n f ( x , y ) , x ) < x. J T h u s j ( j - s u p ( i ( j - i n f ( x , y ) , x ) - x since the inequality x <_ j ( ~ - s u p ( i ( ~ - i n f ( x , y ) , x ) J is also obtained by a) of (2) of Definition 6.1.10 so t h a t (x A~ y) Vj x -- x. F u r t h e r m o r e , by a) of (2) of Definition 6.1.10, y <_ i , - s u p ( x , y ) and, therefore, i
y so t h a t (x Vi y) A~ y -- y. In a similar manner, using
i~:-inf(i(~-sup(x, y), y) -
Definition 6.1.10, we (:an prove the equalities (x V~ y) Aj x - x and (x Ai y) V~ y -- y. Thus condition GL4 is also satisfied and s _ {L, A1, V1, A2, V2} is a G.lattice. (2) Let the algebra s - {L, A1, Vl, A2, V2} be a G.lattice. Assume t h a t y4 Then 4
x .z--->xAiy--y
for each pair (x,y) c ( L i x L i )
U(Lj xLi).
L, --_< are partial orders on Li. Moreover, the binary relation 4
is a
quasi order on L - L 1 U L 2 . Indeed, it is obvious t h a t x 4 x for each x c L. Let us show t h a t x 4 Y and y 4 z imply x 4 z for each elements x, y, z c L. To this end, we shall consider the cases: x, y, z 6 Li;
x 6 Li,
y, z C L j ;
x, y C Li,
z 6 Lj;
and x, z c Li,
T h e case x, y, z c Li is obvious since 4 IL, =<--. i
If x c L~, y, z c L j , t h e n x 4 y ,,+----5, y A ~ x -- x and y 4 z ,z-----~, z A j y -- y.
Therefore x - y Ai x -- (z
t h a t is, x - - z A ~ x ~ x 4 If x, y E L~, z E L j , then
Aj y) Ai x and by GL3, we have z.
x 4 y ,z-~, y A ~ x -- x and y 4 z ,,v----5, z A ~ y -- y.
Therefore x - y Ai x -- (z A~ y) Ai x and by GL3, we have (zAiy) Aix--zAi(yAix)--zAix, t h a t is, x - - z A i x ~ x ~
z.
y c Lj.
208
VI. Generalized Boolean Algebra and Related Problems
Finally, we assume t h a t x, z c Li, y c L j . T h e n x4
y ,,r
y Ai x -- x,
Y4
z ,~-~, z Aj y -- y.
Therefore x - y At x -- (z Ay y) At x and by GL3, we obtain (Z A j y ) A i X - - z Ai
(y At x) - z Ai x
so t h a t x - z At x ~ x 4 z. T h u s the relation 4 is a quasi order on L - L1U L2 such t h a t 4 IL, =_< and s o / : q - (L, 4 ) is a goset. i
Now, assuming t h a t i a - i n f ( x , y) - x A~ y and i(~-sup(x, y) - x Vi y for each pair (x, y) c (Li x L i ) U ( L j x L i ) , let us prove t h a t the g o s e t / : q - (L, 4 ) is a G.lattice in the sense of Definition 6.1.10. First, we show t h a t if x c L j and y E L i , then i~-inf(x, y) - y <--5, i t - s u p ( y , x) - x, t h a t is x Ai y -- y <--> y Vj x -- x. Indeed, by GL4, we have x A i y - - y implies y V j x implies x Ai y -- (y V j X) A i y - - y. Therefore Y4
(xAiy)Vjx-
x and y V j x -
x
x <----~, x A i y = y ,,+--~, y V j x = x.
Now we proceed to considering the fulfilment of (1) and (2) of Definition 6.1.10. Let (x, y) E Li x Li be any pair. Then, according to the a r g u m e n t s t h a t follow Definition 6.1.12, i , - i n f ( x , y) a n d / , - s u p ( x , y) coincide with inf{x, y} and sup{x, y}, respectively, in the usual sense. Moreover, if z c L j and z 4 x, z 4 y, t h e n by GL3, (xA~y) Ajz--xAj(yAjz)--z
and so z 4
xAty~z4
inf{x,y}.
Quite in a similar manner, one can prove t h a t x ~ z Ai y 4 z imply sup{x, y} 4 z. Hence a) is also satisfied and thus (1) of Definition 6.1.10 is completely valid. If ( x , y ) c L j x L~, then by GL4, (x A~ y) V~ y -- y so t h a t x At y _< y and, i
therefore, i~.-inf(x, y) <_ y. Similarly, (x A~ y) Vj x -- x so t h a t x A~ y 4 x, t h a t is, i
i , - i n f ( x , y) 4 x. F u r t h e r m o r e , let z E Li, z _< y and z 4 x. Then, by GL3 and i the definition of 4 , we have z -
y
-
y -
(x
z)
y -
x
A, y)
-
(y
-
y)
so t h a t z < x Ai y and, therefore, z < i~-inf(x, y). Finally, let z E L j and z 4 y, i
z < x. Then, by GL3, J
i
(X A i y ) A j z -- x A j ( y A j z) z x A j z z Z a n d s o
z ~
x
At y,
t h a t is, z ~ i~j-inf(x, y). Using a similar reasoning and taking into account the equivalence y ~ x <---> y Vj x = x, we can prove the fulfilment of the rest of the conditions for the s u p r e m u m case in (2) of Definition 6.1.6. Thus the goset L;q = (L, ~ ) is a G.lattice. (3) T h e gosets L; = (L, ~ ) and (L;a) q have the same basic set L = L1 U L2. Hence to prove t h a t L; = (L;a) q, it suffices only to show t h a t the quasi orders on L; and (L;a) q coincide. First, we shall prove t h a t if the goset/2 = (L, ~ ) is a G.lattice, then y ~ x ~ i c - i n f ( x , y ) = y for each pair (x,y) c (Li x L i ) U ( L j x L~).
6.1. Gosets, Generalized Lattices, ...
209
Indeed, if (z, y) c Li x Li, this equivalence is obvious since ~ IL, ----<. Now, let i
(z, Y) C L j x Li and 9 ~ z. By a) of (2) of Definition 6.1.10, i , - i n f ( z , y) <_ 9 since /
s - (L, 4 ) is a G.lattice. On the other hand, y _< y and y 4 x. Hence, by b) of i
(2) of Definition 6.1.10, y < iG-inf(z, Y) so that y i
i~-inf(z, y).
Conversely, if the goset s - (L, 4 ) is a G.lattice, then by a) of (2) of Definition 6.1.10, y 4 z. Thus, if a goset s - (L, ~ ) is a G.lattice, then y 4 z implies i~;-inf(z, 9) - Y for each pair (z, 9) c (Li x L i ) U ( L j x Li). Therefore, to prove that (s _ s it remains to apply successively (1) and (2) of Theorem 6.1.15. (4) The algebras s - {L, A1, VI,A2, V2} and (s have the same basic set L -- L 1U L 2. Hence to prove that (s _ s it suffices to show that the operations on s and (s coincide by applying successively (2) and (1) of Theorem 6.1.15 since s {L, A1, V l , A2, V2} is a G.lattice. K] Based on Theorem 6.1.15, we denote by s - { L 1 , A 1 , V l , 4 , L 2 , A2, V2} a G.lattice, defined by means of a goset. It is obvious that s - {L1 - {(9, e},A1,VI, d , L 2 -- {19, e},A2, V2}, where A1 -- A2 -- A and V1 - V2 - V are usual lattice operations, is a G.lattice and s is a G.sublattice of any G.lattice t; - {L1, A1, V1, 4 ,L2, A2, V2} such that L1 n L2 - {19, e}. In the sequel we shall consider only G.lattices with the G.zero and G.unit elements, that is, of the type t ; - {L1,A1,V1,19,4 ,e, L2, A2, V2}, where 19, e c L1K1L2.
Our next theorem is most important for further consideration because it shows that the posets, composing a goset, are in a one-to-one correspondence of special type. Theorem
6.1.16. Let s - {L1, A1, Vl, 19, 4 , e, L2, A2, V2} be a G.lattice and ()~,,X~) be a pair such that X, " Li ~ L j are maps, defined as follows" X, (x) - x Vj 19 f o r each x c L~. T h e n the conditions below are satisfied: -
(1)
(x)
-
9
a ,d
(e)
-
e,
-
(2) X, o X, -- idL, so that X, (X, (x)) - x f o r each x c Li; thus X1, X~_ are bijections and X, - X T, 1 (3) X, o Ai -- A j o (X,, X,) a n d X, o V~ - V j o ( X , , X , ) so t h a t
and T h u s x <_ y ~ }(, (x) <_ X, (Y) f o r each pair (x, y) c L~ x L~ i j (the i s o t o n i c i t y of the m a p s X,). (4) }(, o Ai -- Aj o (~,,,}(,) and X, o V~ - Vj o ( X , , X , ) so that
and
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212
VI. Generalized Boolean Algebra and Related Problems
()~1,)~2)
The pair X -
is called the G.identity operator.
c
b
x. ~. x. .~ . I j -~
d
Let (L = L1 U L2, 4) be the goset in Diagram 2, where the quasi order relation x ~ y is determined as for Diagram 1 by the arrangement of elements x, y E L 1 U L 2 = { a , b , c , n } U { c , d , m , n } at different or the same levels.
n
Diagram 2 If we define the binary operations Ak, Vk : Li x x 4 y <--5, x V k y = y for each pair ( x , y ) c verification of axioms L1 - GL4 shows t h a t s = G.lattice with (9 = n, e = c, X1 (a) = m, ~1 (b) --- d ,
Lk ~ Lk as y A~ x = x ,', ;, Li x Lk, then an elementary { L 1 , A 1 , V1, 4 , L 2 , A2, V2} is a
X2 (~'~) :
a
and X2 (d) = b.
P r o p o s i t i o n 6.1.17. For a G.lattice s = {L1, A1, V1, 0, 4 , e , L2, A2, V2}, the following conditions are satisfied:
(1)
(x /~ y) v~ (x /~ z) = x / ~ (y v~ (~ /~ ~)) ., ,.. .'. :, (z ~ x implies (x Ai y) Vi z -- x Ai (y Vi z)) z. ;.
.'. :. (x v~ y) A~ (x v~ z) - x v~ (y A~ (x v~ z)) ,'. ;. ,'
:. (x ~ z implies (x Vi y) Ai z -- x Vi (y Ai z))
if x, y, z E Li or x C L j , y, z c Li.
(2)
(x Aj y) v~ (~ A~ z) = x A~ (y v~ (~ A~ ~)) ,, ,, .',
,~
(z ~ x implies (x Aj y) Vi z = x Ai (y Vi z))
.'
:.
.' :. (x vj y) A~ (x v, z) = x v~ (y A~ (x v~ z)) .'. '. -(
:.
(x 4 z implies (x Vj y) Ai z = x Vi (y Ai z))
if x, y E L j , z 6 Li or x, z E L i , y E Lj. Proof. Since all these conditions are proved in a similar manner, we can do with proving only (2) for x, y c L j , z 6 Li. First, let (xAjy) Vi(xAiz)--xAi(yVi(xAiz))
and z 4
x.
T h e n z = x A~ z and hence
(x Aj y) v~ ~ = (x A~ y) v~ (x A~ z) = x A~ (y v~ (~ A~ ~)) = ~ A~ (y V~ ~). Conversely, let z ~ x imply (x Aj y) V~ z = x A~ (y V~ z). Then x A~ z ~ x implies that
(x Aj y) v~ (x A~ z) = x A~ (y v~ (x A~ z)) and thus the first equivalence is true.
6.1. Gosets, Generalized Lattices, . . .
213
Using a similar reasoning one can prove that the trird equivalence is also true. By virtue of GL3 and (5) of Theorem 6.1.16, it suffices to prove only that (z 4 x implies (x Aj y) Vi z -- x Ai (y Vi z)) ~' ,<---5, (x 4 z implies (x Vj y) Ai z -- x Vi (y Ai z)). If x 4
z, t h e n x V i z - z a n d
(x v j y) A~ ~ - (x v j y ) / ~ (x v~ ~) - ((x v~ ~) nj (y v5 x)) A~ ~ -
= (((~ v~ ~) Aj y) v~ ~) v~ e - ((z Aj y) v~ ~) v~ e -- (x v j ( z / ~ y)) v~ e - x v~ ((z/~j y) v~ e) - x v~ ( y / ~ ~). Conversely, z ~ x implies x Ai z -- z and
(x A~ y) v~ z - (x Aj y) v~ (x A~ ~) - ((x A~ ~) v~ (y Aj ~)) v~ e -
= (((x A~ ~) vj y) Aj x) v~ e - ((z vj y) A~ x) v~ e - (x Aj (~ vj y)) A~ ~ = x / ~ ((~ vj y) n~ ~) - x / ~ (y vj z).
n
D e f i n i t i o n 6.1.18. A G.lattice s - {L1, A1, Vl, O, 4 , e, L2, A2, V2} is said to be modular (or Dedekind) if it satisfies one of the equivalent conditions both in (1) and (2) of Proposition 6.1.17. This notion is closely associated with D e f i n i t i o n 6.1.19. A G.lattice t; - {L1, butive if the following laws hold:
A1, V l , I~, 4 , e,
L2, A2, V2} is distri-
CDL1 x A~ (y V~ z) -- (x A~ y) V~ (x A~ z) and x Vi (y Ai z) -- (x Vi y) Ai (x Vi z) ifx, y, z c L i o r x 6 L j ,
y, z c L i .
x v~ (y A~ z) - (x vj y)A~ (x V~ ~ ) i f x, y C f j , ~ c f~ or x, z c f~, y c f j . G u n 2 (x v~ V) n~ z - (z A~ z) v~ (v A~ z) and (xAiy) Viz--(xViz)Ai(yViz) ifx, y, z E L i o r x E L j , y, z c L i . (xVjy) A~z--(xAiz)Vi(yAiz) and ( x A j y ) Vi z -- (x Vi z) A~ (y Vi z) if x, y C L j , z ~ L i o r x , z~Li, y~Lj. It is obvious that if s -- {L1,A1,V1,Q),4 ,e, L2, A~,V2} is a distributive G.lattice, then {Li, Ai, Vi, (9, e} are distributive lattices in the usual sense and every G.chain is a distributive G.lattice. GDL1 and GDL2 are called the G.distributivity laws. Let us denote by I GDL1, II GDL1, III GDL1, and IV GDL1 the first, second, third, and fourth equalities in GDL1, respectively, and similarly for GDL2. Theorem have:
6.1.20.
For a G.lattice s
IGDL1 ~
IIGDL2,
-
{L1,A1,VI,~,4 ,e, L2, A~,V~}, we
IIGDL1 ~
IGDL2
and
IIIGDL1 <---->, IVGDL2, so that GDL1 ~
IVGDL1 ~
IIIGDL2
GDL2.
Proof. All the proofs essentially utilize Definition 6.1.12 and Theorem 6.1.16.
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220
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
= y Ai ((a Aj x) Vi (x Ai y)) -- (a Aj x) V~ (x Ai y) --
-- (a Aj x) Vi (a Ai y) Vi (x Ai y) and so x = y Vj @. As for the implication (4) ~ (3), it is an immediate consequence of the notion of a G.neutral element and Theorem 6.1.22. D C o r o l l a r y 6.1.25. For a G.lattice 12 = {L1,A1,V1,Q),4 ,e, L2, A2, V2} the following conditions are satisfied: (1) Every G.neutral element is G.standard and coG.standard. (2) Every G.standard (coG.standard) element is (V,A)-distributive ((A, V)-distributive). (3) Every G.standard and c o G . s t a n d a r d or every G.standard and (A, V)-distributive element is G.neutral.
(1) By (3) and (4) of Theorem 6.1.24, every G.neutral element is (V, A)-distributive, (A, V)-distributive, and
Proof.
aA~x=aA~y,
aV~x=aV~y
aAjx=(aAiy)
Aje,
imply x = y
aVjx=(aViy)
for x, y c L ~ ,
V j O imply x = y V j @
for x C Lj, y E L~. Therefore it remains to use (1) .z---->.(2) in Theorem 6.1.24. Assertion (2) follows directly from the equivalence (1) .z----5. (2) in Theorem 6.1.24. (3) If an element a e Li is G.standard and coG.standard, then by (1) <---5, (2) in Theorem 6.1.24, it is (V, A)-distributive, (A, V)-distributive and a Ai x = a Ai y, a Vi x = a Vi y imply x = y for x, y C Li,
aAjx=(aAiy)
Aje,
aVjx=(aViy)
V j O imply x = y V j O
for x E L j , y E L~. Thus it remains to use (3) z---> (4) in Theorem 6.1.24. If an element a c L~ is G.standard and (A, V)-distributive, then by (1) ~ in Theorem 6.1.24, it is (V, A)-distributive and a Ai
X --- a A i y ,
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a Vi X
Aje,
=
a Vi y
(2)
imply x = y for x, y c Li,
(a Vj x) = (a Vi y) Vj (9 imply x = y V j ( 9
for x C Lj, y E Li. Thus it remains to use (3) ,z---->,(4) in Theorem 6.1.24.
[3
D e f i n i t i o n 6.1.26. A G.lattice s = {L1, A1, V1, 0, ~ , e , L2, A2, V2} is said to be G.complemented if there exists a pair ~ = (pl, p2) such that Pi : Li ---, Lj are maps and x Aj ~ ( x ) = @, x Vj ~ ( x ) = e for each element x c L~. The pair p = (~1, ~2) is called a G.complementation operator. P r o p o s i t i o n 6.1.27. For a distributive G.lattice s = {L1, A1, V1, 0, 4 ,e, L2, A2, V2} the G.complernentation is unique.
6.1. Gosets, Generalized Lattices, ...
221
Proof. Let q p ' - (p~, p~) be another pair, where ~{'L~ -+ Lj are maps such that x Aj p~(x) -- O and x Vj p~(x) - e. Then by GDL2 we obtain ~ ( x ) - e v~ ~ ( x ) - (. Aj ~'~(.)) vo ~ ( . ) = (x vj ~ ( . ) )
Aj ( ~ ( ~ ) vj ~ ( . ) )
- ~ Aj (~'~(.) vj ~ ( . ) )
- ~'~(.) vj ~(~)
so that ~{(x) _< ~i(x). The case ~i(x) <_ ~{(x) is proved in a similar manner and thus ~ - ~ .
j
J
D
Now we are ready to introduce the basic notion. D e f i n i t i o n 6.1.28. A G.Boolean algebra (briefly, GBA) is a distributive and G.complemented G.lattice. In the sequel a GBA will be denoted by A - {A1,A1, V I , ~ I , O , ~ , e , A2, A2, V2, ~2}. It is obvious that if A1 - A2 - A, then A1 -- /~2 -- A and V1 -- V2 -- V are usual lattice operations, (~1 - - ( P 2 - - - - is & complementation operation in the usual sense and, therefore, . 4 - {A, A, V , - , O, e} is a Boolean algebra (briefly, BA) in the usual sense. A G.Boolean subalgebra (briefly, GBSA) of a GBA A is a set B c_ A1 U A2, B • A1 - B1 ~ Z ~ B2 - B A A2, which is closed under the four G.Boolean operations: Ai and ~ , or, V~ and pi. It is obvious that any GBSA contains 0 and e. Take place the following obvious statement.
Proposition
6.1.29.
Yo~ ~ y
~o~-~.~pty fa.~ily {B~- B~ u B ~ } ~ of
GBSA's of a GBA A, the intersection Bo -
( ~ B~) U ( ~ Bt2) is a GBSA tET
o/A.
tET
Therefore for any subset D c A1 U A2, there exists a GBSA of At generated by D, which is the smallest GBSA containing D. Clearly, for a GBA A - { A 1 , A 1 , V I , ( ~ I , ~ , 4 ,e, A2, A2, V 2 , ~ 2 } , the subset {a, ~ l ( a ) , ~=},e} U {X1 (a), (~l(a), {~, e} C n l U n 2 together with the corresponding G.lattice operations is a GBSA. E x a m p l e 6.1.30. Let us consider a nonempty BS (X, the binary operations
A~, v~. ( ( ~ , j ) - o ~ ( x ) •
T1
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7"2)
and define
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as follows" U A~ V -
T~int Tj cl U ~ V and U V~ V - ~-~int rj cl(~-~ int T1 cl U U V).
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6.1. Gosets, Generalized Lattices, . . .
225
Now we assume that (x, y) ~ A j x Ai and x ~ y. Then, by Theorem 6.1.15, L2 and GL3, we have x
J(;
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9no ~ ( ~ ) = (~ n , x) n , ~ ( ~ ) = ~ ( ~ ) n , (~ n , ~) = - ( ~ ( ~ ) n~ ~) no ~ = ~.
On the other hand, by (1) above and (3) of Theorem 6.1.31, we have x - y = (9 <----> x Aj ~ i ( y ) = (9 ~ Jc;
q~j(x) Vi y = e
and hence ( p j ( x ) V ~ y ) A j x = x so that by GDL2, ( ~ j ( x ) A j x ) V j ( y A j x ) = y A j x = x. Thus x 4 y. (6) We shall only consider the conditions without brackets. By condition (3) above
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v _< ~,(x) ~ z
< :, x < ~ i ( y ) ~ J
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y <_ ~o(x) ~ i
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by (2) of Theorem 6.1.31,
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On the other hand, F j ( x ) < y, F j ( z ) <_ v imply F j ( x ) Ai qpj(z) < y Ai v i i i
and thus p j ( x Vj z) <_ y Ai v gives (x Vj z, y Ai v) C Pv(~j). i
K]
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6.1. Gosets, Generalized Lattices, . . .
229
(8) By analogy with the proof of (6) of Corollary 6.1.32, for each pair (x, y) c
Aj x Aj, we have j
J
Note now t h a t ~j o @j - ~i o ~j. Indeed, ~j
O ~)j - - ~ j o ~ i O Xj - - Xj
and ~i o q~j - ~i o ~i o Xj - ;~j.
Moreover since (x, y) e PA(~j) <---> Y <_ ~j(x) and ~j are antitone, we have J
~ j ( ~ j ( ~ ) ) <_ ~j(y) ~
~(~j(~))
i
_< ~j(y) ~
(~j(x), ~j(y)) e P ~ ( ~ ) .
i
Therefore
(~j(x), ~ j ( y ) ) e Finally, (x, y) e PA (~j) ~
P~(~)~
(~j(y), ~ j ( x ) )
9P ~ ( ~ ) .
Y < ~j (x) and, by (4) above, J J
(9) We will prove only four implications; the others can be proved similarly. I f ( x , y ) e PA(~j), z 9Li and z 4 y, then by (7) above, y ~ ~j(x) s o t h a t z < ~ j ( x ) s i n c e ~ A~ --<. Therefore (x,z) 9PA(~j).
j
Now, if (x,y)
J 9P v ( ~ j ) , z
9n~ and x 4
z, then we have ~j(x) < y and J
x 4 z. But ~j(x) < y ~ ~ ( y ) < x so t h a t ~ ( y ) ~ z and, by (7) above, we i j have (z,y) 9Pv(~i). The rest can be proved in a similar manner. Assertion (10) is proved in the same way as (7) of Corollary 6.1.32. (11) By (7) above, for each x 9A~, the element ~ ( x ) 9A~ is its complement in the usual sense and so ~i(x) is unique. Hence Xl
-
{A1,
A1, V l , ~)1, O , S , 1
e} and ,.,42 -- {A2, A2, V2, ~2, (~, <_, e} 2
are BA's. To prove t h a t ~, "Ai ~ Aj are isomorphisms, by virtue of (2) and (3) of Theorem 6.1.16, it suffices to prove only t h a t )/, (~i(x)) - ~j(X, (x)) for each x 9Ai. By (5) above ~ ( x ~ (x)) - ~ j ( x v~ e ) - ~ ( x ) / ~
~(e)
- ~(x)/~
~ - x~ ( ~ ( x ) ) .
Furthermore, let a 9Ai be an atom of Ai and X~ (a) 9Aj be not an atom of My. Then there exists an element b 9Aj \ {O} such that b < X, (a) - a Vj (9. We J
have ;~ (b) < ~ ()~ (a)) - a, where X;~(b) # (9 since X~ are isotonic bijections. The contradiction obtained shows t h a t )/, (a) is an atom of Aj. The reverse implication is obvious since )/j ()/~ (a) - a. Proving the second equivalence is clear. (12) First, let (x,y) 9A~ x A~. Then x 4 y ~ x < y <---5. x A ~ y - x. Therefore
x - ~ - x a~ ~ ( ~ ) - (x a~ ~) a~ e~(~) - x a~ (~ a~ e~(~)) - x a~ e - o. i
230
VI. G e n e r a l i z e d Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
Conversely, if x - y i
Oi(xAiOi(y))-
- x/~i ~i(Y) - 0 , t h e n by (2), (3), a n d (4) above, we have
Oi(x)Viy-
e a n d so, by ( 7 ) a b o v e , ( r
Now let (x, y) E Aj x Ai. T h e n x 4 y ~ x - y -
r
i
-
x < y.
y Aj x -- x. T h e r e f o r e
(y Aj
= ((r
y) e Pv(r
-
A, y) Aj
(r
A,
Aj (y Aj x ) ) A,
-
- e.
O n t h e o t h e r hand, if x - y - O, t h e n e i ( x Ai ~i(Y)) -- e j ( x ) Vi y -- e. T h e r e f o r e i
(~j(x), y) c P v ( ~ j ) a n d hence p j ( ~ j ( x ) ) <_ y, t h a t is )/j (x) - x Vi O _< y z----> x 4 y. i
i
(13) We will prove only t h e equivalence in brackets. i
g~i(Y) <_ x Vi (9 ~ ( y , x Vi (9) C P v ( e i ) . i I m m e d i a t e c o n s e q u e n c e of previous r e a s o n i n g is ": :" ~i(Y) 4 x Vi (9 ~
Example
6.1.34.
I-I
C o n t i n u i n g E x a m p l e 6.1.30, let us consider t h e pairs X First, let U e ( 1 , 2 ) - O Z ) ( X ) - 1-OT)(X). T h e n by T h e o r e m 6.1.16 a n d (1) of L e m m a 0.2.1, we have
(X1,X2) a n d ~ - ( ~ 1 , ~ 2 ) .
X1 (U) - U V2 2~ - 7-2 int 7-1 cl (7-2 int 7-1 cl U U 2~) - 7-2 int 7-1 cl U. 9(2, 1)-O~P(X) - 2-OZ)(X), t h e n by (3) of C o r o l l a r y 2.1.7, we have
Now, if U
)/2 (U) - U V1 2~ - 7-1 int 7-2 cl (7-1 int 7-1 cl U U 2~) = 7-1 int 7-2 cl 7-1 int 7-1 cl U - 7-1 int 7-2 cl U. Hence ( 1 ) o f L e m m a 0.2.1 a n d ( 3 ) o f C o r o l l a r y 2.1.7 give )(~2(~1 ( U ) ) -- T2 int --
Wl
T1
cl U
V1 ~
-- T1
int r2
(T 1 int 7-1 cl r2 int
cl
T1
cl U U 2~) -
int r2 el rx int 7-1 el r2 int 71 cl U - 7-1 int r2 cl r2 int 71 cl 72 int r l el U -
-- 71 int r2 cl 72 int 7-1 cl U - 71 int r2 cl 71 int 7-1 cl U - 71 int r2 cl U - U so t h a t X~ (XI ( U ) ) )(~1
()(~2(U))
T1
--
U and int r2 cl U V2 2~ - rg, int r l cl (r2 int
= r2 int r2 cl r2 int r2 cl
T1
T1 C1 T1
int r2 cl U U 2~) -
int r2 cl U - r2 int r2 cl r l i n t r2 cl U =
= r2 int r2 cl 72 int r2 cl U - U, a n d so X I ( X 2 ( U ) ) @I(U)-
U. F u r t h e r m o r e , if U
X2(~I(U))-
~2(T2
9(1, 2 ) - ( 9 / ) ( X ) -
i n t ( X \ U))
-T1
1-O~D(X), t h e n
int 7-2 c172 i n t ( X \ U) -
= r l int r2 cl r2 int rl c l ( X \ U) and, by (3) of C o r o l l a r y 2.1.7, we o b t a i n ~)1 ( g )
- - T1
int 7-2 el T1 int - - 7"1
T1
c l ( X \ U)
- - T1
int ~-2 cl
T1
int ~-2 c l ( X \ U) -
int 7"2 c l ( X \ U) - 7"1 i n t ( X \ U).
6.1. Gosets, Generalized Lattices, . . .
231
Consequently, g A 1 ~)1 ( g )
-
T 1 int r2 cl U ~ T 1 i n t ( X \ U) -
U N T 1 i n t ( X \ U) - 2~,
U Me ~)1 ( g )
-
T 1 int 72 cl (T1 int T1 C1U [_J T1 i n t ( X \ U)) -
= 71 int (72 cl 71 int T 1 cl U U 72 cl 71 i n t ( X \ U)) = 71 int (72 cl U U 72 cl 72 i n t ( X \ U)) = 71 int (72 cl U U 72 cl(X \ 72 cl U)) - X. Now, if U c (2, 1)-OT)(X) - 2-OT)(X), t h e n ~2(U) - X1 (~2(U)) - X1 (71 i n t ( X \ U)) - 72 int 71 cl 71 i n t ( X \ U) = 72 int 71 cl 71 int 72 cl(X \ U) - 72 int 72 cl 72 int 72 cl(X \ U) = 72 int 72 cl(X \ U) - 72 i n t ( X \ U). Therefore U A2 ~2(U) - 72 int 71 C1 g A 7"2 i n t ( X \ U) - U N 72 i n t ( X \ U) - 2~, U V2 ~2(U) - 72 int 7-1 cl (72 int T 1 cl U U 72 i n t ( X \ U)) = 72 int (71 cl ~-2 int 7-1 cl U U T 1 cl 72 i n t ( X \ U)) = 72 int (71 cl U U T 1 c l ( X \ 72 cl U ) ) -
X.
Finally, ~i(~i(U))
- 7i int ( X \ 7i i n t ( X \ U ) ) - 7i int 7~ cl U - U
for each U c ( i , j ) - O D ( X ) -
a
i-07P(X).
c
Let ( A = {A = A l U A 2 = {a, b, c, d, m , n}, 4 } ) b e the goset in D i a g r a m 3 where A1 = { a , b , d , m}, A2 = { b , c , d , n}, and the quasi order relation on A is defined as follows: if x, y c Ai and x is connected with y by the solid line directed upwards from x to y, then x -< y; if x and y are connected by the horizontal broken line, t h e n x -< y and y -< x. Finally, i f x , y c A \ { b , d } are not connected, t h e n x and y are not comparable.
Diagram 3 Let binary operations Ak, V k : A i x Ak ~ Ak be defined as follows: y A i x = x ,z-----+, x ~ y ,z-----~, x V k y = y
for each pair of c o m p a r a b l e elements (x, y) c Ai x Ak and yAix=xAky=d,
yVix=xVky=b
for each pair of n o n - c o m p a r a b l e elements (x, y) c Ai x Ak. T h e n it is not difficult to see t h a t A = { A 1 , A 1 , V l , ~ 1 , 0 , 4 , e , Ae, A2, V 2 , ~ 2 } is a G B A , where (3 = d, e = b, ~1 ( a ) ~-~ ?~, ~1 (T/~) - - C, ~ 2 ( C ) - - T/~, ~2(Tb) = a , ~ ) l ( a ) = ?Tt,
e l ( ? T t ) -- a ,
r
= /t,
e2(/t)
~-- c,
232
vI. Generalized Boolean Algebra and Related Problems
Xl(a)-
c,
xl(m)-n,
x2(n)-m.
X2(c)-a,
Next, for a GBA A - {A1, A1, VI,~I, Q), ~ ,e, A2, A2, V2, ~2}, we shall define two more binary operations >" Ak x Ai ~ Ai in the manner as follows" iG
x-->y-~k(x)
Viy if ( x , y ) ~ A a x A i .
T h e o r e m 6.1.35. For a GBA .4 = { A 1 , A I , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2,~2}, the following statements hold: (1) ~i(x ~
y) = x - y if (x,y) c Ak x Ai.
z(;
(2) ~ - - .
iG
x
iG
JG
x = ~, e - - - .
x = ~, ~ - - .
i(;,
y) ~
(x ~
y) ~
z~
x = x and
kG
ZG
~G
y = x V~ y if (x,y) c Ak x Ai.
(4) x ~ y = e ~ x ~
y/f(x,y)~AkxA~.
zG
Zc;
x
i(;
e = ~(~)
iG
>e=eifxEAi.
(3) (x ~
(5) x ~
9 =x, ~ - . .
i~
y = (~(y) ~
~ ( x ) ) V~ O if (x, y) e A~ x A~ and
3c;
> Y = W(Y) ~
V j ( x ) if ( x , y ) E Aj x A~.
zc;
(6) x /~ (x ~
y) = x / ~ y, y / ~ (x ~
ZG
y) = y and
~G
x ~
(x Ai y) = x ~
y if (x,y) c Ak x Ai.
y ~
(x A~ y) = y ~
x if (x,y) c A~ x A~ and
y ~
(x Ai y) = (y ~
iG
zG
(7) x ~
3~
x ~
iG
3~
~ ( x ) if (x,y) c A~ x A~ and
~(y) = y ~
3~
~i(Y) = (Y ~
JG
x) Vi (3 if (x,y) E Aj x Ai.
V j ( x ) ) Vj (3 if (x, y) c A j x Ai.
z c;
(s)
(x ~
y)/~
zo
(x ~
z) = x
---., (~/~ ~)
zG
iG
if x, y, z ~ A~ or x ~ Aj, y, z ~ A~ and
(~ ~
~) ~ (x ~
3c;
z) = ~ ~
ZG
(~ ~ ~)
iG
if x, y c A j , z c Ai or x, z ~ Ai, y ~ A j . (9)
(x
~
~)/~ (~ ~
zo
z) = (~ v~
~) ---.
zG
z
iG
if x, y, z ~ A~ or x ~ Aj, y, z ~ Ai and
(x ~ zG
z)
~ (~ ~
z) = (x v~ ~) ~
zG
z
iG
if x, y ~ A j , z ~ Ai or x, z ~ Ai, y ~ A j .
(10)
x --.~ (y ~ iG
zG
z) = (x Ai y) ---> z = y ~ iG
~G
(x ~
zG
z)
a~a
~~
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6.1. G o s e t s , G e n e r a l i z e d
Lattices, ...
237
Once again to emphasize the importance of the G.identity operator, recall that to define a GBA above, we first introduced the notion of a goset, then successively the notions of a G.lattice, a distributive G.lattice, a G.complementation operator and, finally, the notion of a GBA. Now that we have fully covered the (fundamental) details, we can introduce the notion of a GBA more easily by using only the well-known notions of the theory of Boolean algebras and the G.identity operator. T h e o r e m 6.1.37. Let ( L - L1 U L2, 4 ) be a goset, where (L~, A~, V~, (9, e) are lattices and L1 UI L2 - {0, e}. If there exists a pair X - (X1, X2) such that X, " Li --+ Lj are maps and X,(x) ~ x ~ X,(x) for each element x c Li, then X, are isomorphisms. Proof. First, let us prove that X, (X, (x)) - x for each x c Li. Clearly,
x, (x, (~)) v x,, (x) ~ x, (x,, (*)) and, therefore,
so that x _< X, (X, (x)) and X, (X, (x)) < x i
since 4
i
Li --~" Hence X, (X, (x)) - x, X, are bijections and X,
-
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i
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X,,(x) and X,(Y) 4 Y 4 X~(Y)
imply
x~ (x) Aj x, (y) ~ x A~ y ~ x, (x) Aj x, (v). On the other hand since
we obtain that and so because ~ In, --<. J Finally, the case of Vi can be proved similarly and thus X~ are isomorphisms. F1 As in the first case the pair X - (X1, X2), is called the G.identity operator. C o r o l l a r y 6.1.38. Let (L - L1U L2, ~ , X) be a goset, where (L~, A~, V~, 0, e) be lattices, L1 N L2 - {O,e} and X - (X1, X2) be a G.identity operator. Define binary operations
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• L~)u (L, • L , ) - ~ L~
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240
vi. Generalized Boolean Algebra and Related Problems
Conversely, if Z: is a G.lattice, then (Ls, Ai, Vs, O,e) are lattices in the usual sense, satisfying the conditions which correspond to a) and b) of (4). Therefore (Li, Ai, Vi) are Boolean algebras ([26, [29]) and thus, by (c) of Corollary 6.1.38, s is a GBA. D The pair I = (11, 12) is called the G.rejection operator. Our next result is concerned with a V-formation and the corresponding strong a m a l g a m a t i o n in the sense of [127]. Let s = {L1, AI, V1, 4 , L 2 , A2, V2} be a G.lattice. We define the binary relation (c) on L1 U L2 as follows: if x c Li, y c L 1 U L2, then x(c)y ~
y = x or y = x Vj (~.
Let us prove t h a t (c) is the congruence o n L1 U L2. In the first place, note t h a t (c) is the equivalence relation o n L 1 U L 2. (1) Suppose that x, xl, y, yl E L i and x(c)y, Xl(C)yl. Then x = y and XI --- Yl. Hence (x Ai X l ) ( c ) ( y Ai Yl) and (x Vi X l ) ( c ) ( y Vi Yl) since X Ai Xl = y Ai Yl a n d x Vi Xl = y Vi Yl.
(2) If x, Xl E Li, y, Yl E Lj, then y = x Vj I~ -- x Aj e, yl = Xl Vj (~ = Xl Aj e and we have
y Aj yl - (x Aj ~) Aj (Xl Aj ~) - (x A~ x,) Aj ~, y vj yl - (x vr e) vj (x~ vj e) - (x v~ Xl) vr e since/2 is a G.lattice so that
(x Ai Xl)(e)(y Ai Yl) and (x Vi xl)(c)(y Vj Yl). (3) If x, y C Li, Xl, Yl C L j , then x - y, Xl - yl and it follows that
(X Aj Xl)(C)(y Aj Yl), (x Vj Xl)(c)(y Vj Yl). (4) If x, yl 6 Li, X l , y C Lj, then y - x Aj e, Yl -- Xl Ai e and x Aj Xl - (x Aj ~) A~ Xl - y Aj Xl - y Aj ((~1 A~ ~) A~ ~) -
= y A j (Yl Aj e) -- (yA i Yl) Aj e. Similarly, x Vj X 1 -- (y V i Yl) Vj (~1, that is,
(x Aj Xl)(c)(y Aj Yl) and (x Vj Xl)(c)(y Vj Yl). Thus (c) is the congruence on L1 U L2. The congruence class, containing an element x E L1 U L2, is denoted by Ix], while the set of classes [x] - by s Let us define binary operations 9, U ' ( s x s ~ s as follows: [x] N [y] -- [x As y] and [x] U [y] - [x Vi y] for each pair (x,y) E L k x L i . It is not difficult to see that ( s [(9], [e]) is a lattice in the usual sense, obtained from s by sticking each element x c Li \ {(~, e} to the element x Vj (~ and the elements (9, e to themselves. Moreover, the maps
mi " (Ls, Ai, Vi, (9, e) ~ (C/c, n, I I, [0], [el), defined as m s ( x ) -
[x], are isomorphisms.
6.1. Gosets, G e n e r a l i z e d L a t t i c e s , . . .
241
Since the lattice C(2) = {L = {(~, e}, A, V} is a sublattice of any lattice which contains (9 and e, we obtain t h a t if s = { L 1 , A 1 , V I , ( ~ , 4 ,e, L2, A2, V2} is a G.lattice, then the quintuplet (s tl,t2), where t~ : s --, L~ are algebra embeddings, is the V-formation since m l O t l = m 2 o r 2 , t h a t is, the diagram ml
L1
s is commutative. Moreover, (ml, m2, C/c) since
t
this
>C/c
t2 > L2 V-formation
is strongly a m a l g a m a t e d
m , ( L 1 ) F1 m2(L2) - m l ( t l ( L ( 2 ) ) - m2t2(L(2))
by
[127].
Our consideration in this section will not be complete unless we make at least a casual mention of the real-valued functions on G B A ' s and bitopological GBA's. For this, we need to introduce a few handful notions. Suppose s {LI,A1,V1,Q},4 ,e, L2, A2, V2} is a G.lattice. Then the elements x E L1 U L2 and y c Li are disjoint, t h a t is, x d , y if x Ai y -- ~ . Furthermore, an element x 6 L1 U L2 is disjoint from a set E C_ L 1 U L 2 , t h a t is, x d c E , if x Ai y -- 0 an for each element y C E~, where Ei - E N Li. D e f i n i t i o n 6 . 1 . 4 0 . Let A = {A1, A 1 , V I , ~ I , Q } , ~ ,e, A2, A2, V2,~2} be a G B A and E = E1 U E2 c A1 U A2. T h e n E is said to be a G.disjoint system of elements of A if xd~((Ei \ { x } ) U Ej), where x c Ei is an arbitrary element, t h a t is, if x Ai y = (3 for each y c E~ \ {x} and x Aj z = (9 for each z c Ej. The notion of a G.disjoint system of elements is used to give the following D e f i n i t i o n 6 . 1 . 4 1 . Let f i [ - {A1, A 1 , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2, q~2} be a GBA. A pair p (#1,#2), w h e r e p i " Ai ~ R are finite maps, is said to be a G.quasi measure on A, if pi(x) >_ 0 for each element x c Ai and
xEE 3
yEEi
xEEj
yCE,
for every finite G.disjoint system E = E1 U E2 C A1 U A2. It is obvious t h a t if p = (pl, p2) is a G.quasi measure, then #~(0) = 0. P r o p o s i t i o n 6 . 1 . 4 2 . If p = (#1,#2) is a G.quasi measure on a G B A A, then x 4 y implies that pk(x) <_ #~(y) for each pair (x, y) ~ Ak x A~.
Pro@ First, let x, y E A~ and x < y. T h e n the elements x and y - x = y Ai ~i(x) i
are disjoint, y - x Vi ( y - x) and hence i
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6.2. Generalized Ideals and their Variety . . . .
243
i f x 4 Y, then Cli(x) 4 Clj(y) and Inti(x) 4 Inty(y) for each pair (x, y) c Ai x Aj.
(9)
x = Cl~(x) ~
~i(x) = Inti(~i(x)) z--> ~i(x) = I n t j ( p i ( x ) )
and x = Inti(x) ,z--->, ~i(x) = Cli(~i(x)) ,z----5,~i(x) = Clj(~i(x)) for each x c A i. The proof, based on elementary calculations, is omitted. A more detailed development of the questions connected to bitopological GBA's and quasi measures on GBA's, in our view, is of independent interest also.
6.2. Generalized Ideals and their Variety. Stone Family of P r i m e Generalized Ideals All our further constructions are essentially connected with the notion, introduced by
Definition 6.2.1. A G.ideal (briefly, GI) of a G.lattice/2 = {L1, A1, Vl, 1~, 4 , e, L2, A2, V2} is a pair I = (I1,/2), where Ii C_ Li and which satisfies the following conditions: (1) I f x c I l U / 2 andycIi, thenxV, ycIi. (2) If x E I1 U/2, y E L1 U L 2 and y 4 x, then y c I1 U 12. E x a m p l e 6.2.2. Let A = {A1,A1,V1,q21,(~,~,e, A2, A2, V2,~2} be a GBA and p = (p l, p2) be a G.quasi measure on A. Then it is not ditticult to see that I = ( I I , h ) , where I~ = {x c A , : p~(x) = 0}, is a GI. Moreover, note that the pairs ({a,d}, {c, d}) and ({m, d}, {n, d}) in Diagram 3 are GI's. Since for every GBA A = {A1, A1, V1, qP1, (~, 4 ,e, A2, A2, V2, P2}, the system { A 1 , A 1 , V I , ( ~ , 4 ,e, A2, A2, V2} is a G.lattice, in the sequel we shall consider, in general, GBA's.
P r o p o s i t i o n 6.2.3. Let A = {A1, A1,V1,~91,O,~ ,e, A2, A2, V2,~2} be a GBA. Then a pair I = (/1,/2), where I~ c_ A~, is a GI if and only if {I~, A1, V1, 4 , /2, A2, V2} is a G.sublattice of the G.lattice {A1, A1, V1, 0 , 4 , e, A2, A2, V2} and x 6 I1 U 12, y 6 A~ imply x A~ y E Ii. Proof. First, let I=(I1, h ) be a GI and let us prove that {/1, A 1 , V I , ~ , / 2 , A2, V2} is a G.sublattice of the G.lattice {A1,A1, V1,0, 4 ,e, A2, A2, V2}. Indeed, if x c I1 U/2 and y c Ii, then by (1) of Definition 6.2.1, z Vi y c Ii. It is evident that z Ai y 4 z and by (2) of Definition 6.2.1, x Ai y E I1 U/2 ~ x Ai y 6 I/. Thus {/1, A1, V1, 4 ,I2, A2, V2} is a G.sublattice. Now, if x c I1 U 12 and y c A~, then x Ai y 4 z and once more applying (2) of Definition 6.2.1 gives that z Ai y C I 1 U / 2 ,z-----N,x Ai y 6 I~.
244
VI. Generalized Boolean Algebra and Related Problems
Conversely, let {I1, A1, V1, ~ , I2, A2, V2} be a G.sublattice of the G.lattice { A 1 , A 1 , V 1 , O , 4 ,e, A2, A2, V2} and x E I1 U / 2 , y C Ai imply x Ai y C Ii. Let us prove t h a t the conditions (1) and (2) of Definition 6.2.1 are satisfied. Indeed, if x c I1 U / 2 and y E Ii, t h e n x Vi y c Ii since {I1,A1, V1, 4 ,I2, A2, V2} is a G.sublattice, t h a t is, (1) of Definition 6.2.1 is satisfied. Finally, if x c I1 U / 2 , y E A1 U A2 and y ~ x, t h e n if, for example, we consider the case y c A j , we obtain t h a t y - x A j y and, therefore, y c lj c I1U/2, t h a t is, (2) of Definition 6.2.1 is also satisfied. D It is obvious t h a t if I - (I1, I2) is a GI, then Ii are ideals in the usual sense and the pair I - (A1, A2) is a GI. It is likewise obvious t h a t for a GI I - (I1, I2), we have x E I1 U 12, y C Ii ~ x V i y C Ii. M o r e o v e r , I1 7s A1 ~ 12 ~= A2 and, therefore, a GI I - (I1,I2) is said to be proper if I~ ~ A~. Thus, by (2) of Definition 6.2.1, I - (I1, I2) is proper
e---->, e g I1 <-->, e c / 2 .
Hence, using (1) of the latter definition, the a r g u m e n t s between T h e o r e m 6.1.31 and Corollary 6.1.32, and (7) of Corollary 6.1.33, we obtain: I -- (I1,/2) is proper -'
<---5, P v ( ~ l ) N (11 x / 2 ) - 2~ ~' :,
,'- Pv(~2) K~(I2 x I1) - 2~ ~ -'
P v ( ~ l ) K I (I1 • I1) - 2~ ,:
:,
:- Pv(~b2)A (I2 x I2) - 2~.
Note further t h a t I ~ - ({(9}, {@}) is called the zero GI, and for any proper GI (I1, I2), we have 11AI2 - {(9}. It is not difficult to verify how I1 a n d / 2 are i n t e r c o n n e c t e d in any proper GI. Following (6) of T h e o r e m 6.1.16, for each x c Ai, we have x 4 x Vj 1~ and x Vj 1~ ~ X. Now by virtue of (2) of Definition 6.2.1, we obtain x c I~ ,<--->, x Vj (~ E Ij. Therefore any proper GI can be w r i t t e n as I - (I1 - { x } , I 2 - { x V2 (9}), where Ii C Ai so t h a t 1 I I 1 - /21. Moreover, if A1 contains an a t o m a (<---5, A2 contains the a t o m a V2 (~), then it is clear t h a t I - ({a, @}, {a V2 (9, @}) is s t r u c t u r a l l y the simplest type of a GI after the zero GI I ~ Let I - ( I I , h ) and I ' - ( I ~ , I ~ ) be any two GI's o f a G B A A. Then, by virtue of the above reasoning, I
-
I1
--
I~ <-->, I2 -- 1s and I1 C I~ <-->, h C I;.
In the former case we write I - I', while in the l a t t e r - I < I'. Hence I < I ' ,' :, (I < I ' or I - I'). Clearly, the binary relation < is a partial order on the set Z{/t - (I~,/~) " t e T} of all GI's of the G B A A. T h e zero of this poser coincides with the zero GI I ~ and the unity - with the unit GI I e - ( A 1 , A 2 ) . As usual, a subfamily {/t - (I~, I~)" t c To c T} of 2- is a chain i f / t <_/t, o r / t , <_/t for each pair t, t ~ c To. Proposition any intersection
6.2.4. For a GBA A- {A1,A1,VI,~I,0,~ ,e, A2, A2, V2,~2} o f G I ' s a n d the u n i o n o f a n y c h a i n i f G I ' s is a GI.
P r o o f . First, let { I t - ( I f , I t ~ ) " t c To c T} be any family of GI's of a G B A A. We are to prove t h a t I - (I1,/2) - ( ~ I~, ~ I t) is also a GI. Assume t h a t tETo
tETo
6.2. Generalized Ideals and their Variety . . . .
245
x 6 I1 U /2 and y 9Ii. If x 9Ii, then x ~ I t for each t ~ To and, by (1) of Definition 6.2.1, x V i y 9 I tfor e a c h t 9To so t h a t x V i y 9Ii. I f x 9Ij, then x 9 I tfor each t ~ To and, therefore, x V~ y 9 I tfor each t 9To, t h a t is, x V~ y ~ I~. Before we proceed to proving (2) of Definition 6.2.1 note t h a t
( N
( N z:) - N
t E To
t E To
t E To
t E To
t E To
o
Indeed, it is clear that
t E T~~
Hence it can be assumed t h a t x c {@, e} and x c
n (I~ u I~). In that case x c I~ t6To
or 2; C I~ for each t E T O since A1 n A2 -
{E:),e} and, t h e r e f o r e , 2; c
n
I~
tGTo
or x C n
I2t. Hence x c ( n
t 6 To
I1t) u ( n
t 6 To
I2t) 9Now assume t h a t x c I1 U I2,
t 6 To
y 6 A1 U A2 and y ~ x. W i t h o u t loss of generality let us consider the case where X 9I1 -- n I~. Then y ~ x and x 9 I~for each t c To imply x 9 I~ U I2t for each t6Tc) t 9To and y ~ x. By (2) of Definition 6.2.1, y 9 I~ UI2t for each t 9To so that y C n (1~ u/2t) -- ( n 1~) u ( n I ~ ) - I l U I 2 . t E Tu
t E Tu
t E Tu
Next, let us assume t h a t {It - (I~,It2) " t 9To C T} is a chain of GI's a n d I -- (11,22) -- ( U I~, U I~). T h e n x c I x u / 2 a n d y 9 I i i m p l y x c
( U zf)u( U zl) t E To
If x
t E To
9
t E To
t E To
.ayc U If. t 6 To
U It, then there exists an index tl
9To such t h a t x c I tx. Similarly,
t6To
there exists an index t2 c To such t h a t y 9 t2. I It is clear t h a t I~ 1 c I~ 2 or I t~ C_ I tl . W i t h o u t loss of generality let I tl C I: 2 . Then x 9 I~ ~and, therefore, x V~ y 9I~ 2 C Ii. Now assume t h a t x 9 U I~. Then there exists t l e To such t h a t y 9 I~ ~ tE Tu
sinceyE
U It and there existst2 t E To
9
such t h a t x
9
~ sincex
9 U It. It is t E To
obvious t h a t / t ~ <_/t2 o r / t 2 _
246
VI. Generalized Boolean A l g e b r a and R e l a t e d P r o b l e m s
of a GBA A and
z'-
{I~, - (I~', I~tt ).
t'
c To c T, (B,, B~) _< I~, for ~,~h t' ~ To }
then 27' r 2~ as I c - (A1, A2) E Z'. The smallest (with respect to <_) GI, containing (B1, B2), is said to be generated by the pair (B1,B2). Clearly, the GI, generated by the pair (~, ~), is the zero GI. Also, note that {27, A, V} is a lattice in the usual sense with I ~ and U as the zero and the unit element, respectively, where for I ' - (I[, I~), I " - (I~', I~') E Z, we have
I' A •
n I~', I~ n I~')
and I ' V I " is the GI, generated by the pair (I{ O I{', I~ O I~'). Before proceeding to our next important theorem, let us recall that by (6) of Theorem 6.1.16 if x, y E A~, z E Aj, then X ~ y, i
X ~ Z ,,~,---)', X \ / j (9 ~ y,
X\/j(9
~ Z. j
Hence, taking into account the fact t h a t for each GI I - (I1,I2), we have Ij X, (Ii), the following equivalences are obvious: (For each element x C I1, there exist finite sequences of elements al, a2,. 99an c A1, b l , b 2 , . . . , bm E A2 such t h a t x <_ al Vl a2 Vl "--Vl an and x 4 bl V2 b2 V2 1 ..9 V2 b,~.) .z---> (For each element x E I1, there exist finite sequences of elements al,a2,...,an c A1, b l , b 2 , . . . , b m C A2 such that x V 2 ( 3 <_ bl V 2 b 2 V 2 " " V 2 b m and 2 x V2 O 4 al V1 a2 Vl .-. V1 an.) ~ (For each element x c I1, there exist finite sequences of elements bl, b2, . . . , bm E A2, al, a2, . . . , an C A1 such t h a t x _< (bl Vl 1
(9)vI(b2VI(9)VI" "VI(DrnVI(9) a n d x
4 (al V2(9)V2(a2V2(9)V2. . .V2(anV2(9).) ." "," (For each element x E I1, there exist finite sequences of elements bl, b 2 , . . . , b~ c A2, al, a 2 , . . . , an e A , such that x V~ (3 < (al V2 (9) V2 (a2 V2 (9) V 2 " " V2 (an V9 (9) 2
and x V~ (9 4 (bl V~ (9) Vl (be V~ (9) V l - . . Vl (b,~ Vl (9).) Note t h a t by virtue of GL3 we can easily ascertain by induction that the equalities (a 1 V 2 e ) V 2 (a 2 V 2 1~) V 2 " "
V2 (a n V2 l~) -- (a 1 V 1 a 2 V l . . .
V 1 an) V 2 (~)
and
(hi Vl e) v, (b~ v, e) v , . . . v, (bm V, e) - (hi v~ b~ v~... v~ bin) v, e are valid for finite sequences of elements al, a 2 , . . . , an 6 A1, bl, b 2 , . . . , bm 6 42. T h e o r e m 6.2.5. L e t A {A1, A 1 , V 1 , ~ 1 , 0 , 4 , e , A2, A2, V2,~2} be a G B A and (B1,B2) be a pair such that Bi C A i . T h e n a pair I - (I1,I2) is a GI, w h e n e v e r it satisfies one of the following equivalent conditions: (a) I1 - {x E AI" there exist finite sequences of elements al, a 2 . . . , an C B1 and b l , b 2 , . . . ,bin E B2 such that x < al V1 a2 V 1 . . . V1 an, x ~ bl V2 1
b2 V 2 " " V2 bin}.
6.2. G e n e r a l i z e d Ideals and their Variety . . . .
247
(b) 12 - {y c A2" there exist finite sequences of elements c l , c 2 , . . . , c k and d l , d 2 , . . . ,dz E B2 such that y < dl V2d2 V2 . . . V2dl, y ~ Cl V1 2 -9. V 1 Ck}. Moreover, this CI I - (I1,/2) is generated by the pair (B1, B2) if and only each element a ~ Bi, there exists a finite sequence of elements b,, b 2 , . . . , bt such that a 4 bl V j be V j . . . V j bt.
c B1 c2 V1 if f o r c Bj
P r o @ First, we shall see t h a t I - (I1, h ) is a GI, t h a t is, (1) and (2) of Definition 6.2.1 are fulfilled. Indeed" (1) Clearly, we have the following four variants: y E I1, x c I1 or x E /2; y C 12, x C I1 o r x C 12. Since all these variants are proved by a similar scheme, we shall consider only the case x c I~, y ~ / 2 . By (a) and (b), there are finite sequences of elements a l , a 2 , . . . , a n ; C l , C 2 , . . . , c a ~ B1, b l , b 2 , . . . , b ~ ; d l , d 2 , . . . , d z ~ B2 such t h a t
x <_ al V1 a2 V 1 - . . V1 an,
x 4 bl V2 b2 V 2 " " V2 bm
y_
y% ClVlC2Vl...VlCk.
,
and 2
Therefore 2
and Let el al,e2 a2,...,en b , , . . . , g,~ - b,~,gm+l - d l , . . . , elements e l , e 2 , . . . , e n - b k C B 1 virtue of which x V2 y c I2. (2) Let x c I1 U I2, y c A1 -
-
-
-
x c I1,
-
y c A,
-
an, en+l Cl,...,en+k ck and gl bl,g2 g,~+z - dz. It is clear t h a t the finite sequences of and g l , g 2 , . . . , g m + l C B2 are those sequences by -
-
-
-
-
-
-
-
U A2 and y d x. Here we also have four variants:
or y c A2;
x c I2,
y c A1 or y c A2.
We will prove only the case where x c I2, y E A1 and y d x; the others can be proved similarly. By (a), there exist finite sequences of elements a l , a 2 , . . . , a , ~ c B1, b l , b 2 , . . . , b m E B2 such t h a t x < bl V2 b2 V2 . . . V2 bm and x d al V1 a2 V1 ..- V1 an. 2
It is obvious t h a t y _~ al V, a2 V1 . . . VlaN and y ~ b, V2 b2 V2 .-. V2 b~ 1
so t h a t y E I1. Thus I = (I1,/2) is a GI. Now let us prove the second part of the theorem. If the GI I = (I1,/2), constructed above, is generated by the pair (B1, B2), then Bi C_ Ii and, therefore, the "only if" part is obvious. Conversely, let there exist for each element a c B~, a finite sequence of elements bl, b 2 , . . . , bt c By such t h a t a ~ bl Vj b2 Vj . . . Vj bt. Then it is clear t h a t a E Ii and, therefore, Bi C_ Ii. Hence, it suffices to prove
248
VI. Generalized Boolean Algebra and Related Problems
that I - ( I i , h ) is a smallest GI such that ( B I , B 2 ) <_ I ,e---->, Bi c_ Ii. Indeed, if I ' - (I~, I~) is any GI for which (B1,B2) <_ I' and x E h is any element, then by assumption, there exists a finite sequence of elements a l, a 2 , . . . , an C Bi such that x <_ al Vi a2 Vi . . . Vi an. Clearly, a l , a 2 , . . . , a n C B i C_ I~ implies x E I~ so i that Ii c_ I ' because x E Ii is an arbitrary element and, therefore, I _< I'. D This theorem gives rise to the following interesting C o r o l l a r y 6.2.6. Let A - {A1, A1, V I , ~ I , 0 , 4 , e , A2, A2, V2, qD2} be a GBA and let I - (11, I2) be the GI generated by a pair (B1,B2), where Bi c Ai. Then the following conditions are equivalent: (1) I - (I1, I2) is nonproper. (2) There is an element a E B1 for which there ezist finite sequences of elements a l , a 2 , . . . , a n E B1, b l , b 2 , . . . , b , ~ E B2 such t h a t p l ( a ) ~ al V1 a2 Vl -.- Vl an and ~ l ( a ) _< bl V2 b2 V2 ... V2 b,~. 2
(3) There is an element b E B2 for which there exist finite sequences of elements C l , C 2 , . . . , c k c B1, d l , d 2 , . . . , d l c B2 such that ~2(b) ~ dl V2 d2 V2 . . . V2 dz and ~2(b) <_ cl Vl c2 V 1 . . . Vl ck. 1
(4) There is an element c E B1 for which there exist finite sequences of elements e l , e 2 , . . . , e t E B1, g l , g 2 , . . . , g r C B2 such that ~bl(C ) 4 gl V2 g2 V2"'" V2 gr a n d ~bl(C ) ~ el V1 e2 V I " " V1 et. 1
(5) There is an element d E B2 for which there exist finite sequences of elements P l , P 2 , . . . , P u c B1, q l , q 2 , . . . , q v E B2 such that r 4 Pl ~/1 P2 V1 . " VlPu and r < ql V2 q2 V2 ... V9 qv. 2
Proof. Applying Theorem 6.2.5, in four cases of the existence of finite sequences of the elements of B1 and B2, we respectively obtain ~ l ( a ) c I2, ~2(b) c 11, r E I l a n d ~ b 2 ( d ) c / 2 . T h e r e f o r e a , c c B1 C_ I1, b, d E B2 C_ /2 and (1) of Definition 6.2.1 imply that a V 2 q ~ l ( a ) - - e e I 2 , bV1 ~ 2 ( b ) - e e I 1 ,
cV1 ~ b l ( c ) - e e I 1 , and d V 2 ~ b 2 ( d ) - e e I 9
so that I - I e - (A1, A2) in each one of the four cases. Conversely, let I - I ~ - (A1,A2). Then e c I1 - A1, e E / 2 - A2. Hence, by virtue of Theorem 6.2.5, there exist finite sequences of elements al, a 2 , . . . , an E B1, bl, b2,... ,bin E B2 such that e - al V1 a2 Vl . . . V1 an -- bl V2 b2 V2 "" V2 bin. Clearly, for each element a E B1 and each element b E B2, we have ~ l ( a ) 4 al V1 a2 V 1 . . . V1 an, ~l(a) < bl V9 b2 V2-.. V2 bm; 2
~2(b) 4 bl V2 b2 V2.-. V2 bin, g)2(b) <_ al V2 a2 V 2 . . . V2 an; 1
~bl(a) 4 b~ V2 52 V 2 . ' - V 2 bin, ~bl(a) _< al V1 a2 V I ' " V1 an; 1
@2(b) 4 al V1 a2 V I " " V1 an, ~2(b) _< bl V2 b2 V2"'" V2 bin. 2
~]
Thus for the GI I - (I~, I2), generated by the pair (B1, B~), to be nonproper it is sufficient that there exists at least one element of the set B1, satisfying the
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252
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
and
{xv =
{y ~ A j "
e.
xe
there exists an element y~ ~ I~ such that y ~ z V~ y~},
is a GI. Moreover, this GI is generated by the GI I' - (I~,I~) and the element z 6 Ai \ I~ (so that I - (I1,I2) is the smallest GI for which I' < I and z E Ii) if and only if for each element y' c I}, there exists an element x E I~ such that y'4x. Pro@ First we are to prove t h a t I - (I1,/2) is a GI. Since in our concrete case z E A~ \ I ' in proving (1) of Definition 6.2.1 we shall consider b o t h y E I~ and ycIj. If y E Ii, z c I1 U/2, t h a t is, if x E I i or x E Ij, then in b o t h cases, we obtain __ z Vi y < z Vi (x' Vi y'), where x' Vi y' E I~ since x', yl E I il and I iI is an ideal in i the usual sense. Therefore, by condition, x V i y E I i . On the other hand, if y E I o, x E I~ or z c Ij, then in a similar manner, we obtain x Vj y 4 z V~ (x' V~ y'), where z' Vi y' E I~, and, therefore, x Vj y c Ij. Let us prove (2) of Definition 6.2.1. For this we assume t h a t x E I1 U / 2 , y 6 A1 U A2 and y 4 x. We shall consider only the cases x E Ii, y c Aj and z E Ij, y E Ai since the other cases are proved quite similarly. For x c Ii and y 6 Aj, we obtain y 4 z Vi z', while for x E Ij, y E Ai, we obtain y _< z Vi z', i
where x' c I[. Therefore y c Ii. Now, let us prove the second part. If a GI I - (I1,/2) is generated by a GI I ' - (I~ , I~) and an element z E Ai \ I '~, then I ' < I and, by the first part, for each element y' c I5 there exists an element y" E I~ such t h a t y' 4 z Vi y". Hence the conditions I~ c Ii and z E Ii imply z V~ y" c Ii, t h a t is, z - z Vi y" is the required element. Conversely, let us assume t h a t I - (I1,I2) is the GI, constructed using the GI I ' - (I~, I~) and the element z E Ai \ I~. It is clear t h a t I~ c Ii and z c Ii. If y' c I}, then the existence of an element x c Ii with the condition y' 4 z implies y' E Ij and, therefore, I~ c Ij since y' is an a r b i t r a r y element of I5. Thus I ' < I and it remains only to prove t h a t if I " - (I{', I~') is any GI such t h a t I ' < I " , and z c I~', then I _< I " . Indeed, for an a r b i t r a r y element x E Ii there is an element x~EI~suchthatx<_zVix ~ ButI ~
x E I~~. Similarly, if y c Ij, then there is an element y' c I~ such t h a t y 4 z Vi y'. Since z Vi y~ C I "i , we obtain y E I~t and so I _< I". D Note that, by virtue the first part of T h e o r e m 6.2.7, for the GI I - (I1,/2), we can write the structure of Ij in more precise terms as follows:
{xv -
{y E Aj - {y c A j "
e-
xc I,} -
9there exists an element y' E I~ such t h a t y 4 z Vi y'} there exists an element t' E I} such t h a t y <_ z V j t'}, J
where obviously t ~ - y' Vj 0 .
6.2. Generalized Ideals and their Variety . . . .
253
C o r o l l a r y 6.2.8. Let A = { A 1 , A I , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2,~2} be a GBA, I = (I1, I2) be the GI, generated by a GI I ' = (I~,I~) and an element z ~ Ai \ I~. Then the following equivalences hold:
I = (h, h)
e Ij
9
Proof. Since, by (1) of Corollary 6.1.33, ~ = ~j o X,, (2) of Corollary 6.1.32 and (2) of Theorem 6.1.16 imply X, o X:, o ~ i = X, o ~Pi ~
9~i = X, o ~'i,
and, hence, if x c Ai is any element and I = (I1,/2) is any GI, then c
x,
c 6.
Thus it remains to prove the first of our equivalences. If ~ ( z ) c I}, then I' < I and z r imply that I = I e = (A1, A2). Conversely, let I = I e = (A1,A2). Then there exists an element x I E I~ such that e = z Vi x'. Therefore ~i(e) = ~i(z) Aj ~i(x') = O and by (5) of Corollary 6.1.32, ~ i ( z ) 4 x'. Thus ~ i ( z ) c I~ since x' c I~. rq Now, we have come to the most important variety of GI's which will be essentially used in our further reasoning. A GI I = (I1,I2) of a GBA is said to be maximal if it is proper and has no property to be contained in a proper GI of A so t h a t there does not exist a proper GI I ' - ( I ~ , I~) such t h a t I < I'. D e f i n i t i o n 6.2.9. A GI I = (/1, I2) of a GBA .4 = {A1, A1, V1, ~ 1 , (~, 4 , e, A2, A2, V2, ~2} is said to be prime if it is proper and the following condition is satisfied: if x E A1 u A2, y 6 Ai and x Ai y C Ii, then either x r I1 U/2 or y r Ii (or both x c 11 U 12 and y c / ~ ) . Therefore, if I = (I1,I2) is a prime GI and x c Ai is any element, then x e Ii or pi(x) e Ij(..v---> ~i(x) = X, (pi(x)) e Ii), but not both, because the GI I = (I1,/2) is proper. On the other hand, if there exists an element x c Ai such that x c I~ and V)i(x) c Ij, then x Aj ~ ( x ) = (9 C Ij and to obtain a contradiction, it suffices to apply the condition of Definition 6.2.9. T h e o r e m 6.2.10. Let I = ([1,12) be a GI of a GBA A = {A1, A1, Vl, (/91, (~), ~---~, e, A2, A2, V2, ~2}. Then the conditions below are equivalent: (1) I = (I1, I2) is a prime GI. (2) For every element x c Ai either x E Ii or pi(x) E I j ( < - - > r x. c both
=
Pro@ The implication (1) ----5, (2) is shown in the discussion preceding Theorem 6.2.10. (2) ~ (1). By (2) it is obvious that the GI I = (I1,/2) is proper. Now assume t h a t x, y c A~, x A~ y C/~ and x ~ 5 , YE I~. Then by (2), ~i(x), ~ ( y ) c Ij and by ( 1 ) o f Definition 6.2.1 and ( 2 ) o f Theorem 6.1.31, we have F~(x)Vj F ~ ( y ) = p~(x A~ y) ~ Ij. Therefore by (2), x A~ y r I~, which is impossible. The case x ~ Aj,
254
VI. Generalized Boolean Algebra and Related Problems
y 9A~ and x A~ y 9I~ is proved in a similar manner, taking into account (1) of Definition 6.2.1 and (3) of Theorem 6.1.31. (1) - - > (3). Let the prime GI I = (I~,I2) be not maximal. Then there is a proper GI I ' - (I~,I~) such t h a t I < I'. I f x 9I ~ \ I i is any element, then (1) <---> (2) implies t h a t ~ ( x ) 9 Ij so t h a t ~ ( x ) 9 I5and, therefore, P v ( ~ ) N (I~ • I5) ~ ;~, which contradicts the condition t h a t I' - ( I { , I ~ ) is a proper GI. Hence I = (I1,I2) is maximal. (3) ---->. (1). Let x 9A1UA~, y 9A~ and x A ~ y 9I~. W i t h o u t loss of generality assume t h a t x 9A~ \ I~ and consider the GI I ' - (I~, I~), generated by I - (I1, I2) and x-gI~. Clearly, I ' = I ~ = ( A I , A 2 ) since I = (I1,I2) is maximal. Therefore for e 9Ai, there exists an element z 9I~ such t h a t e = x V~ z. Hence
sincez
9
andyAiz
If we now assume t h a t x 9A j \ Ij and I ' - (I[, I~) is generated by I - (I1, I2) and x - g I j , then I' - I ~ - ( A 1 , A 2 ) and thus for e 9A j , there exists an element z 9Ij such t h a t e - x Vj Z. Hence, by GDL2, we have
y - (x v j z)
y - (x
y)
(z
y) e
sincez 9 a n d z A i y 4 z. Therefore x g [1 U/2 and x Ai y 9Ii imply y 9Ii. Further, assume t h a t x A~ y 9I~ and y-gI~. First, let x 9A~. If I ' - (I~,I~) is generated by I = (I1,I2) and y g / ~ , then I ' = I e = (A1,A2) and, by analogy with the above, for e 9A~ there exists an element z 9Ii such t h a t e = y Vi z. Therefore Similarly, if x 9A j , then I ' = I e = (A1, A2), where I ' is generated by I = (I1, I2) and y g Ii. Hence, by the remark between Theorem 6.2.7 and Corollary 6.2.8, there is an element z 9Ii such t h a t e = y Vj (Z Vj {~). But by the G.commutativity laws, x Ai y 9Ii .e----->.(x Ai y) Vj ~) = (y Aj x) 9Ij and, therefore, x sincez
Aj
9
9 - (v
z) Aj x - (y Aj x) v j (z Aj x)
zimplyzAjx
9Ij
9
P r o p o s i t i o n 6.2.11. For a GBA A1 - {A1, A1, V1, ~1, ~, ~, e, A2, A2, V2, p2} any GI I - (I1, I2) is p r i m e if and only if the goset (A1 U A2, ~ ) is a G.chain. Proof. Let any GI of A be prime and, for example, there are elements m E A1, y E A2 such t h a t x ~ y is false and y ~ , is false. Then by Theorem 6.2.12 and Corollary 6.2.13 below there are prime GI's I x - ( I ~ , I ~ ) and Iy - ( I ~ , I ~ ) such t h a t x g I ~ , y c I~ a n d y g I ~ , x E I~. S i n c e x A 2 y ~ x, x A 2 y < y, by (2) of 2
Definition 6.2.1, x A 2 y I x Cq I v. Then x A2 y Let (A1 tO A2, 4 ) xEAltOA2, yEA~.
e I~ and x A 2 y e I y. Let I - ( I i , I u ) - ( I ~ A I ~ , I ~ A I ~ ) -C I2, but x g I1 and y g / 2 , which is impossible. be a G.chain, I (I1,I2) be a GI and x Ai y C Ii, where Thenx4 yory4 x. If y 4 x, t h e n x A ~ y - y E I ~ , a n d i f
6.2. Generalized Ideals and their Variety . . . .
255
x 4 y where, for example, x 9Aj, then x Ai y 9Ii implies y Aj x -- (x Ai y) Aj e -x 9Ij so that in both cases I - (11, I2) is prime. V] Our next theorem underlies many further constructions. T h e o r e m 6.2.12. For a GBA A - {A1, A1, Vl, ~1, (~, ~ , e, A2, A2, V2, ~2} and each element x 9Aj \ {(9}, there exists a prime GI I* - (I~,I~) such that
Proof. Let A4j - {I - (11,/2)} be a family of all GI's such that x - ~ I j and ~j(x) 9Ii(~=> ~j(x) 9Ij), where x 9Aj \ {(9} is an arbitrarily fixed element. Clearly, A4j ~ ~ since, in particular, the right principal GI (~I(X))I(R)--(II--{ycA
1 " y ~ ~l(X)}, I2-{zcA
2
9 z ~(~l(X)}) 2
9
and the left principal GI
(992(x))i(L)-(Ii-{yCAl
" y ~
1
992(x)}, I 2 - { z
9
" z 4 ~2(x)}) 9
for x 6 A1 \ {(~} and x 6 A2 \ {(~}, respectively. Following Proposition 6.2.4, every chain in 3dj has an upper bound in A4j since the family Adj is partially ordered by the relation <. Therefore by Zorn's lemma, M j has a maximal element I* - (I~, I~). Let us prove that I* - (Ii~, I:~) is the required prime GI. Clearly, x - c I ] and ~ j ( X ) 9 I[, that is, ~)j(X) 9 I~. By virtue of Theorem 6.2.10, it suffices to prove that I* - (I~, I~) is maximal. Let I' - (I~,I~) be a GI such that I* < I'. It is obvious that ~j(x) 9 I~(~' ",, @(x) 9Ij). Also, x 9Ij since if x-E I j, then I' c A4j, which is impossible by the definition of the family 34j and the definition of I* as its maximal element. Therefore P v ( ~ j ) n I} x I~ r ~ so that I ' - I e - ( A 1 , A 2 ) and thus I* - (I~,I~) is a maximal, that is, a prime GI. D C o r o l l a r y 6.2.13. For a GBA A - {A1, A1, V1, 7~1, O, ~ , e, A2, A2, V2, P2} the conditions below are equivalent:
(1) (2)
For each element x c A~ \ {@}, there exists a prime GI such that x-c I~. For each pair (x, y) E (A~ x Aj) \ PA(P~), there exists a prime GI I = (I1,12) such that x -c I~ and y -~ Ij. (3) For each pair (x, y) c Ai x Aj, where x ~ y is false, there exists a prime GI I - (/1, I2) such that x-~ I~ and y E Ij. (4) For each pair (x, y) c (A~ x A~) \ PA(~P~), there exists a prime GI I = (I1, I2) such that x -c Ii and y -c Ii. (s) For each pair (x, y) c A, x A~, where x < y is false, there exists a prime GI such that x-E Ii and y c I~.
i
Pro@ (1) ~ (2). Let (x, y) c (Ai x Aj) \ PA(~i) be any pair. Then by (6) of Corollary 6.1.32, (y,x) c (Aj x Ai) \ PA(~j) and, therefore, y Ai x C Ai \ {@}. By (1), there exists a prime GI I - (I1,I2) such that y Ai x-~Ii and thus, by (2) of Definition 6.2.1, x c Ii and y -c Ij.
256
VI. Generalized Boolean Algebra and Related Problems
(2) ~ (3). Let (x, y) E A~ x Aj be a pair such that x 4 Y is false. Then by (5) of Corollary 6.1.32,
x A~ 9~j(Y) r 0 and (x Ai 9~j(Y), e)
9(Ai x Aj) \ PA(9~i).
Therefore, by (2), there exists a prime GI I - (I1,/2) such that x Ai ~j(y)-EIi so that ~i(x Ai ~j(y)) -- p~(x)Vj y 9Ij since I - ( I i , h ) is a prime GI. Hence 9~(x), y 9Ij and x gI~. (3) --->. (4). Let (x,y) 9(Ai x A i ) \ P~(gi) be any pair. Then, by (7) of Corollary 6.1.33, y ~ 9~(x) is false and, therefore, by (3) above, there exists a prime GI I - (I1,/2) such that y ~ / ~ and 9~(x) 9Ij so that x c/~. (4) ~ (5). Let (x,y) 9A~ xA~ b e a p a i r such that x_< y is false. Then, i
by (12) of Corollary 6.1.33, x A~ r r ~ and by (7) of the same corollary, we have (x A~ r e) 9(A~ x A~) \ PA(~). Thus by (4), there exists a prime GI I - (I1, Is) such that x Ai @i(Y)C Ii so that @i(x A~ t~i(Y)) -- t~i(x) Vi y
9Ii
since I - (I~,/2) is a prime GI. Hence ~bi(x), y 9I~ and x~I~. (5) ~ (1). Let x 9Ai \ {O} be any element. Then x _< ~Pi(x) is false and by (5), there exists a prime GI I - (I1, Is) such that x-~I~.
i
[3
C o r o l l a r y 6.2.14. I f { I t - (I~,I~)}tcT is a family of all prime GI's of a GBA then ( ~ I~, r] i ~ ) - ( { e I , { e } ) .
.A-{A,,A1,V1,9~,,O,~,e, A2, A2, V2,9~2},
tET
tCT
Proof. The proof is an immediate consequence of Theorem 6.2.12.
[3
Thus the intersection of all prime GI's of a GBA is the zero GI. However, we shall see that every infinite GBA (i.e., A1 and, therefore, As too is infinite) has families of prime GI's the intersection of which is the zero GI and which do not contain all the prime GI's of ,4. The following definition will be of much use. D e f i n i t i o n 6.2.15. A family S - {I - (I1,I2)} of prime GI's of a GBA A - {A1, A1, V1,9~a, O, 4 , e, A2, Ae, V2, 9~2} is called a Stone family of prime GI's if ( A t 1 , AI2 ) -- ({(~}, ( ( ~ } ) -- I O. IEs
T h e o r e m 6.2.16. Let S be a family ofprime GI's of a GBA A = {A1, A1, V1, 9~1,O, ~ , e, A2, A2, V2,9~2}. Then the conditions below are equivalent:
(1) S is a Stone family. (2) For each element x E A~ \ {(9}, there exists a prime GI I such that x -~ I~. (3) For each pair (x,y) c (Ai x Aj) \ PA(9~i), there exists a (I1, I2) E S such that x -E I~ and y -EIj. (4) For each pair (x, y) c A~ x Aj, where x ~ y is false, there GI I = (I1, I2) c S such that x-~ I~ and y E Ij. (5) For each pair (x, y) E (Ai x Ai) \ PA(~i), there exists a (I1, I2) c S such that x -~ Ii and y -~ Ii.
= (I1, I2) E S
prime GI I = exists a prime prime GI I =
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258
VI. Generalized Boolean Algebra and Related Problems (7) (8) (9) (10)
/ f x = i a - s u p B , then ~ ( x ) = j a - s u p C d a B for any subset B C A1UA2. C d c B is a GI for any subset B c A1 U A2. Cdaax(L) = qPl(a)I(R) if a E A1, and CdaaI(R) = ~2(a)i(L) if a E A2. IflC = {B} is a non-empty family of subsets of A1UA2, then ~ C d a B =
Cd~
U
B61C
B61C
B.
(11) x = / c - s u p B if and only if x E Cd(;(Cdc~B ) and ~ ( x ) ~ C d a B for any
subset B c A1 U A2. Proof. (1) and (2) are immediate consequences of the respective definitions. (3) If x c u c ( B ) N A i , then b <_ x ~ ~ ( x ) A ~ b - O if b E Bi, and
i
b4 x ~ ~ ( x ) Aj b - (~ if b c Bj so that 7)~(x) c CdaB. Conversely, if x E Aj and 7)j(x) E CdaB, then 7)j(x) A~ b - (~ ~ bEB~,andT)j(x)Ajb-(~~b<_xifbEBj. T h u s x C u a ( B ). J (4) By Definition 6.2.18, we have
Cdc;(Cd~B) - {x c A1
U A2
b 4 x if
" xdGCdGB}
and, therefore, if b E Bi is any element, then x Ai b - O ~
b Ak x -- O for each element x E Cd(;B and so, we have
bdGCdaB ~
b E C d a ( C d c B ).
(5) First, let us prove that if B - B1 U B2 has an i c - s u p B , then for each element y c A1 U A2, we have:
yAiia-SUp(BlUB2)--ia-SUp((
U ( y A I a ) ) U ( U (yA2b))). aEB1
Indeed, let y c Aj be any element.
b6Bg,
< y A i i a - s u p B for each i element x c Bi and y Aj z ~ y Ai ia-sup B for each element z E Bj. Therefore
ia-stlp ( (
U (yAla))U
(
a6B1
Then y A i x
U (yA2 b ) ) ) ~ y A i i a - s l l p ( B 1 U B 2 ) . b6B2
Now, let
u
u( u
aEB1
bEB2
be any element. Then for each x E Bi, we have
x <_ x v~ ~,(y) - (y v~ ~j(y)) A~ (x v~ ~,(y)) - (y A~ x) V~ ~j(y) <_ z V~ ~,(y).
i
i
Similarly, for each x E Bj, we obtain x ~ z Vi ~j(y). Therefore
ic-sup(B 1 U B2) _< z Vi r
i
and y Ai iG-sup(B1 U B2) <
i
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260
VI. Generalized Boolean Algebra and Related Problems
(11) Let x = i c - s u p B . By (5), ydcx for each y E CdoB. Therefore y A i x = (9 implies x Ak y = (9 for each y C C d c B and so x E C d o ( C d c B ). Further, by (7), pi(x) = jc;-sup B and, by (5), qpi(x) Ak y = 19 for each y E C d o ( C d o B ) . But by (4), B C C d c ( C d o B ) so that ~i(x)Akz = (9 for each z C B. Hence ~i(x) c C d c B . Conversely, if ~i(x) c CdoB, then by (3), we have x E u o ( B ). If z c u o ( B ) is any element, then applying once more (3) gives that ~k(z) E C d c B , that is, x At ~k(z) = (9 since x C C d o ( C d c B ). Therefore x 4 z and thus x = i t - s u p B since z c u c (B) is an arbitrary element. [-1 D e f i n i t i o n 6.2.20. Let A = { A 1 , A 1 , V I , q O l , ( 9 , ~ ,e, A2, A2, V2,~2} be a GBA and B = B1 O B2 c A1 U A2. Then B is said to be a G.component of A if B = C d o ( C d c B ) , that is, if C d c ( C d ~ B ) c B. Now we can formulate the following important statements. T h e o r e m 6.2.21. For a GBA .4 the conditions below are satisfied:
=
,e, A2, A2, V2, p2},
{A1,A1,VI,~I,O,~
(1) Every G. component is a GI. (2) Left and right principal GI's are G.components. (3) C d a B is a G.component for every subset B = B1 U B2 c A1 U A2.
(4) If
= {E}
a
family of a. ompo
t , th n ('1 E EC tC
al o
a G.component and thus for every subset B = B1 U B2 c A1 U A2 there exists a smallest G.component A B, containing B. (5) AB = C d a ( C d c B ) for every subset B = g 1 U B 2 C A1 u A2 and, hence, AB is a GI. Proof. Assertion (1) follows directly from ( 8 ) o f Theorem 6.2.19. (2) By (9) of Theorem 6.2.19, CdGa~(c) = pl(a)~(n), and hence Cdc;(Cdc. a l ( L ) ) = C d c ( g p l ( a ) l ( R ) )
= ~92(gPl(a))i(L) = de(L)
if a c A1. Similarly, Cdc(Cdoal(R)
) = Cdc(g)2(a))l(L) = ~l(~2(a))l(R)
= al(R)
ifaEA2. Thus we can conclude that the converse of (1) above is true for the left and right principal GI's. (3) By ( 4 ) o f Theorem 6.2.19, C d o B c C d a ( C d a ( C d o B ) ) . On the other hand since, by (4) of Theorem 6.2.19, B C C d c ( C d c B ) , it follows from (1) of the same theorem that Cd~.(Cd~(Cd~B)) c Cd~B. (4) If E0 = N E, then by (10) of Theorem 6.2.19, EC/C
Zo- [') cd (cd z)ECtC
U CdGZ
EC1C
and it remains to use (3). (5) By (4) of Theorem 6.2.19, the set C d a ( C d a B ) is a G.component, containing B, and hence AB C C d a ( C d a B ) . On the other hand, by (1) of Theorem 6.2.19,
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262
VI. Generalized Boolean Algebra and Related Problems
Note t h a t the pairs ({m, b}, {n, b}) and ({a, b), {c, b}) in Diagram 3 are GF's. In the sequel we shall consider GBA's. It is evident t h a t if F = (Fx,F2) is a GF, then Fi are filters in the usual sense and the pair F = (A1,A2) is a GF. It is likewise obvious t h a t for a GF F = (F1,F2), we have x c F1 U F2, y C Fi ~ x A i y E Fi. Furthermore, it is clear t h a t F1 r A1 ,z----5, F2 r A2, and hence a GF F = (F1, F2) is said to be proper if Fi :/: Ai. Therefore, by (2) of Definition 6.3.1, a GF F = (F1, F2) is proper ,z----5, (~ c F1 ,z----5, (9 c F2 and thus, taking into account (1) of Definition 6.3.1 and using the arguments between T h e o r e m 6.1.31 and Corollary 6.2.32, as well as (7) of Corollary 6.1.33 of the same theorem, we find t h a t F - (F1,F2) is a proper *', )" P A ( ~ 2 ) ( ]
GF <--5. P A ( ~ I ) A
(/!72 x F 1 ) - ~ ~ ",
( F 1 x/t72) -- Z ,(
PA(~)I)N (F 1 x F1)-
Z .'
,','.
" PA (~P2) C~ (F2 x F 2 ) - 2~.
Note t h a t the GF F ~ - ({e}, {e}) is called the unit GF, and for any proper G F F - (F1, F2), we have F1 • F2 - {e}. By (2) of Definition 6.3.1, it is clear t h a t every proper GF has the form F - (F1 - {z}, F2 - {x V2 O}), where x r (3, and hence Fll - IF2[. Let F - (F1, F2) and F ' - (F~, F~) be any two G F ' s of a G B A A. T h e n by virtue of the above arguments F1 - F[ ~ F2 - F~ and F1 c F~ z--->, F2 c F~. In the former case we write F - F ~ and in the latter c a s e - F < F ~. Therefore F < F ~ ,z--5, ( F - F ~ or F < F~). It is obvious t h a t the relation _< is a partial order on the set ~ {Ft - (F~,F~) 9 t cT} of all GF's of the G B A A. The zero element of this poset is the unit GF, while the unit element is the zero GF F - F O - (A1,A2). As usual, a family {Ft - (F~, F~)" t E To c T} is a chain if Ft <_ Ft, or Ft, <_ Ft for each pair of indices t, t ~ c To. A G F F - (F1, F2) of a G B A .4 is said to be maximal or a G.ultrafilter if it is proper and has no property to be contained in a proper GF of A so that, there is no proper GF F ' - (F~, F~) of .4 such t h a t F < F'. Definition 6.3.2. A G F F - (F1, F2) of a G B A .4 - {A1, A1, V1, ~1, ~ , 4 , e, A2, A2, V2, ~2} is said to be prime if it is proper and the following condition is satisfied: if x E A1 U A2, y E Ai and x Vi y E Fi, then either x E F1 t2 F2 or y c Fi (or both x c F1 t2 F~ and y c Fi).
By analogy with the reasoning after Definition 6.2.9, the notion of a prime GF immediately implies t h a t if x c A~ is any element, then x c Fi or ~ ( x ) c Fi(< :. ~2~(x) - Xj(~i(x)) E I~), but not both since the GF F - (F1, F2) is proper. An especially useful assertion is T h e o r e m 6.3.3. For a G B A A - {A1,A1, VI, qPl, ~), 4 ,e, A2, A2, V2, ~2} the following condition (the G.duality) holds: I - (I1, h ) is a GI of A if and only if F-(~2(12),~1(/1))
i8 a G F of,A.
Pro@ It suffices to prove only the implication from left to right since the proof in the opposite direction can be carried out by the same scheme, taking into account the equality ~ i ( ~ j ( I j ) ) - Ij.
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264
VI. Generalized Boolean Algebra and Related Problems
(7) Let F = (F1, F2) be the GF generated by a pair (B1, B2), where Bi C Ai. Then the following conditions are equivalent: (a) F = (F1, F2) is nonproper. (b) There is an element a c B1 for which there exist finite sequences of elements al,a2,. .. ,an E B1 and bl,b~, . . . ,bin ~ B2 such that al A1 a2 A1 "" 9 A1 a n ~ ~ l ( a ) and bl A2 bz A 2 " " A2 bm < ~ 1 ( a ) 2
(c) There is an element b c B2 for which there exist finite sequences of elements C1~ C2~ 9 9Ck9 ~E B1 and d l , d 2 , . . . ,dz E B2 such that dl A2 d2 A2 ".9 A2 dz ~ ~2(b) and C l A l C 2 A l . . . A l C k <_ ~2(b). 1
(d) There is an element c C B 1 for w h i c h there exist finite sequences of elements e l , e2 ~ 9 9e t 9 E ~ B1 and gl, g2, 9 , g~ 9 9E B2 such that gl A2 g2 A2 " ' " A2 g r ~ 1~1 (C) and e l A1 e2 A1 " ' " A1 et 1/21(C) 9 1
(e) There is an element d E B2 for which there exist finite sequences of elements P l , P 2 , . . . ,Pu E 81 and ql,q2,... ,qv E B2 such that pl A1 P2 A1 " ' " A1 P u ~ ~ 2 ( d ) and q l A2 q2 A 2 " " A2 qv ~ 1/)2(d).
(8)
If a E A1, b E A2 and
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81
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--
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andb~ x},F2-{yEA2"
the
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b < y a n d a ~ y}) 2
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(FI-
{xcAI"
a<_x}, F 2 - { y c y c A 2 "
aV20
1
2
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there exists a finite sequence of elements a~,a~2,... ,a fk c B1 such that a~ A1 a~ A I " " A1 a~ ~ y}) and if B 1 - - ~ , ~ ~ B 2 C A2, then the GF, generated by the pair (~,B2), has the form: F = (F1 = {x c A1 : there exists a finite sequence of elements b~l,b~,..., b~ E B2 such that b~l A2 b~ A 2 . . . A2 b~m ~ x}, F2 - {y E A2 " there exists a finite sequence of elements bl, b2,..., bz c B2 such that bl A2 b2 A2 ... A2 bz _< y}).
(10)
2
{a}, B2 = z (B1 = z , B2 = {a}), where a E Aa (a E A2) is an arbitrary element, then the CF, generated by the pair ({a}, ~) ((~, {a})), is called the left (right) principal GF and has the form
I f B 1 ~-
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--
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{ a V2 y "
a 4 x}, F2 - {y E A2" x C A 1 } , F2 - {a V2 y"
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266
VI. Generalized Boolean Algebra and Related Problems
(16) Let F -
(F1,F2) be the GF generated by a GF F ' element z E Ai \ F~. Then F-
(F1,/72)
is nonproper
~
~i(z) c I~ ~
(F~,F~) and an
~i(z) c 1~.
(17) I f .T -
{ F - (F~,F2) 9 F is a GF of a GBA A}, then {U,A,V} is a lattice in the usual sense with F ~ and F ~ as the zero and unit elements, respectively, where f o r F ' - ( F ~ , F ~ ) and F " - ( F ~ ' , F ~ ' ) , we the pair (F~ U F[', F~ U F~').
Proof. Conditions (1)-(17) are immediate consequences of the G.duality and the corresponding statements for GI's. D
T h e o r e m 6.3.5. Let A - {A1, A1, V l , 991, 1~, ~ , e, A2, A2, V2, ~2} be a GBA, ( I i , h ) be a GI, F - (F1,F2) be a GF and I n F - 2J. Then there exists a prime GI I* - (I~, I~ ) such that I <_ I* and I* n F - ;~. I-
Proof. Let 3,t - { I ' - (I~, I ; ) " I < I ' and I ' n F - ~}. It is clear that 34 r ;~ since I E A/t, and Ad is partially ordered by _<. If C - {It - ( I ~ , I t ) " It c All} is any chain in AA, then by Proposition 6.2.4, I " - (UI~, U/t) is a GI. Therefore, by Zorn's lemma, A/l has a maximal element I* - (Ii~, I~). Clearly, I _< I* and I* N F - ;~ so that, it suffices to prove only that I* is prime. Contrary: if I* is not prime, then there exist elements a E (A1 U A2) \ (Ii~ U I~) and b E Ai \ I[ such that a Ai b E I~*. Without loss of generality let us suppose that a c A1 \ I~ and b c A2 \ I~. Since I* is a maximal GI having the property I* n F - ;~, we have a I * n F r 2~ r I*b N F and, therefore, a I[ N Fi r ;g r I[b N Fi. Let, for example, x c aIi~ n F1 and y C I~ b n/7'2. By the definition of aI* and I *b, there exist elements m E I i ~, p c Ii ~ such that x-mVla,
y-pVpb
and m V l a E F 1 ,
pV2bEF2.
Thus t - (m V~ a) A2 (p V2 b) E F2 since F is a GF. By I GDL1 and III GDL2, we have: t
--
=
((?Tt
Vl
a) A2 p)
v)
V2 ((m V1 a) A~ b) p) t,) (a -
t,).
By condition aA2b C I~. Since I* is a GI, it is clear that m A 2 p E I~; also aA2p <_ p 2
andpcIi ~implyaA2pCI~,mA2b4 mandmcIi ~ i m p l y m A 2 b C I ~ . Hence t ~ I~ and thus I* n F ~r ;~, which is impossible, and, consequently, I* - (I~, I~) is the required prime GI. V] C o r o l l a r y 6.3.6. Let I - (I1,/2) be a GI of a GBA A - {A~, A1, V1, Pl, ~, 4 , e, A~, A~, V2, ~ } and a E A~ \ I~. Then there exists a prime GI I* - (I~, I~ ) such that I <_ I* and a-r I [ . P r o @ Without loss of generality let a ~ A2 \ I~ and let us consider the GF aF(R) -- ( F 1 - - { x ~ A l "
a 4 x } , Fg, - {y E A2 " a <_ y } ) .
Then I n a~(~) -- ~ and it remains to use Theorem 6.3.5.
2
6.3. Generalized Filters and their Variety. . . .
267
C o r o l l a r y 6.3.7. Let A - {A1, A1, VI,~I, 1~),~ , e , A ~ , A 2 , V~, ~ } be a GBA, a, b ~ A1 ~ A2 and a r b. Then there exists a prime GI which contains exactly one of the elements a and b. Proof. Let us consider the following cases: (1) a, b ~ A~, a r b. Then for a, b E A1 (a, b c A2) and a <1 b (a < b), it
suffices to consider the GI aZ(L) and the GF bF(c) (respectively, the GI ai(R) and the GF bF(R)) since ai(c) ~ bF(c) -- ~ (ai(R) C~br(R) -- 2~). (2) a, b ~ A~, a r b and a, b are not comparable by 4 IA, = < - If, for example,
i
a, b ~ As, then we also have ai(~) ~ b~(R) - 2~ - a~(n) N hi(n). Indeed, if we consider a <_
a 4
2
and
({:;c C AI" :;c 4 b}, {y c A2" y < b}), 2 then aF(R) A bl(R) r 2~ implies t h a t there exists an element y c A2 such t h a t a _< y _< b and so a < b, which is impossible. The proof for ai(R) and bF(R) is 2 2 2 similar. (3) a E A1, b E As and a -< b. Let us consider bi(R)-
aI(L)-
({X C AI" a: < a}, {g E A2" y ~ a } ) 1
and bF(R) -- ({X ~ n l " b 4 x}, {y ~ n 2
9 b ~y}). 2
It is clear that aI(L) N bF(R) -- 2~. (4) a E A2, b c A1 and a, b are not comparable by 4 . In this case we also have aI(R)NbF(L) -- ~ -- aF(R)Nbt(L). Indeed, let, for example, aI(R)C~bF(R) r ~. Then there exists y ~ A2 such t h a t b 4 9 < a, t h a t is, b -< a, which is impossible, ffl
2
C o r o l l a r y 6.3.8. A n y GI I - (I1,12) of a GBA A - {A~, A1, V1, ~1, (~), ~ , e, A2, A2, V2, ~2} is the intersection of all prime GI 's, which contain I. Proof. Let M-
{It - (I~,I~) " I <_ It, t c T }
and I ' -
(~I~,~')I2t). tET
tCT
Let us prove t h a t I ' - I. Contrary" I' r I and so I < I'. Then there exists a E I~ \ I1 and by Corollary 6.3.6 there is a prime GI I " such that I < I " and a g I~'. But then a g I~ since I ' <_ I". D T h e o r e m 6.3.9. Let A = { A 1 , A 1 , V I , g ) I , 0 , ~ , e , A2, A2, V2, q~2} be a GBA, I = (I1,/2) be a GI and F = (FI,F2) be a GF. If I N F = (I1 N F1, I2 N F2) r ~, then {I1 to Fj, A1, V1, ~ , / 2 tO F2, A2, V2} is a G.convex G.sublattice of the G.lattice {A1, A1, V1, 0, ~ ,e, A2, A2, V2} and, conversely, any G.convex G.sublattice of {A1, A1, V1, (~, ~, e, A2, A2, V2} can be represented in a unique manner as a nonernpty intersection of a GI and a GF.
268
VI. Generalized Boolean Algebra and Related Problems
Proof. First, let I n F = (11 n F1,/2 N F2) r Q and let us prove that {I1 U F1, A1, V1, 4 , / 2 U F2, A2, Vg.} be a G.sublattice of the G.lattice {A1, A1, V1, ( 9 , 4 , e , Ag~,A2, V~.}. L e t a E ( I 1 N F 1 ) U ( h N F 2 ) a n d b E I i N F i . IfaEIjnFj, then by (1) of Definition 6.2.1, a V~ b c I~. On the other hand, a c Fj and a 4 a V i b . Hence, by (2) of Definition 6.3.1, a V i b E F l U F 2 4---> a V i b E Fi and, therefore, a Vi b E I i N Fi. Furthermore, a E Fj, b E Fi and by (1) of Definition 6.3.1, a A~ b E F~. But we also have a c Ij and a A~ b 4 a. Therefore by (2) of Definition 6.2.1, a Ai b E I~ U/2 e-----F,a Ai b c I~ and thus a A~ b E I~ n F~. Now, let us prove that the set (I1N F1)U (I2 N F2) is G.convex. We will prove only the cases, where aEIinFi,
bcIjnFj,
ccAi
and a < c 4 i
b;
the others can be proved similarly. By (2) of Definition 6.2.1, we have c E I1 U /2 .e-----F. c c Ii and by (2) of Definition 6.3.1, c c F1 U F2 .e----F. c E Fi. Thus c E Ii N Fi implies c C (11 n f l ) U (I2 n/72) so that (I1 n F1) U (I1 n F2)
is G.convex and the first part is proved. By Proposition 6.2.4, any intersection of GI's is a GI. Let us prove that this fact is also true for G.convex GI's. Suppose that {It - (I~, I~)" t c T} is a family of G.convex GI's, that is, I~ U I~ is a G.convex set for each t E T, such that
I-- (/1- n I ~ , / 2 tET
-
N tET
and a, b E I1 U 12, c 6 A1 U A2, a 4 c 4 b. Let us consider only the case, where a c I1, b c / 2 and c c A1. Then a c I~, b c I~ for each t c T and since It - (I~,It) is G.convex, c E (I~ U I t) n A 1 ,g-----5,C E I~ for each t c T. Therefore c E I1 and so c c I1 U/2. Hence the set I1 U I2 is G.convex, that is, I - (I1,12) is G.convex. Suppose that s - {L1,A1, V1, ~ ,L2, A2, V2} is a G.convex G.sublattice of { A 1 , A 1 , V 1 , O , 4 , e , A2, A2, V2}. If I - (I1,I2) and F - ( F 1 , F2) are respectively the GI and the GF, generated by (L1,L2), then Li c_ Ii n Fi. By Theorem 6.2.5 and (6) of Corollary 6.3.4, x c Ii N Fi implies that there exist elements b E Lj, c c Li such that c _< x 4 b. Therefore x E L i because L 1 U L 2 is G.convex and /
thus L i - Ii n F~. Finally, let us prove that this representation is unique. Suppose that there exist I' - (I~, I;) and F ' - (F~, F~) such that L 1 U L 2 - (I~ n f ; ) U (I; n F~). It is clear that Ii _C I ' as Li c_ I ' and I is generated by (L1,L2). Similarly, F~ C_ F'. On the other hand, let a c I ' and b E Lj be any elements. Then a V 3 b c I~ and a Vj b _> b E F~. Following (2) of Definition 6.3.1, a Vj b E F~ and, therefore, J a V j b E I } N F ~ . Furthermore, a 4 a V j b a n d L j - I~NF~ implyaVjb EIj. Therefore, by (2) of Definition 6.2.1, a 4 a Vj b implies a c Ii and thus I - I'. Similarly, one can prove that F - F ' . [2
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274
VI. Generalized Boolean Algebra and Related Problems
The rest can be proved by means of (1) of Theorem 6.1.35 in the manner as follows: if (x, y) E Ai • Ai, then y) -
h
(
j(x h
y)) -
h
(
j(x A j
(v) -
h
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h
y) (x)
Similarly, one can prove that
h~(x = ~ y) - hi(x) ~(~
.-ff~' h~(y) for (x, y) c Aj z d~.
~(~,.
(6) This statement follows immediately from Definitions 6.4.1, 6.4.6, and (1)(3) above. (7) If h - (hi, h2)" A ~ A', then by Definition 6.4.1, and (3) above, hi and G h2 are homomorphisms. Therefore the first equivalence is the well established fact. Now assume that (hl(x),h2(y)) e PA(~I) and h - ( h i , h 2 ) " ,4 ~ A' is a G G.isomorphism. Therefore hi and h2 are bijections, and (hl(x),h2(y)) e PA(~I) gives hi(x) A~ ~ ( h 2 ( y ) ) - ~t so that hi(x) A~ hl(~2(y)) - ~t and so hl(x A1 ~2(Y)) - O'. Hence x A1 P2(Y) -- O, that is, (x,y) E PA(~I). Similarly, one can prove that (h2(z), hi(v)) c PA(P~) implies (z, v) E PA(P2). Conversely, let (hl(x),h2(y)) c PA(~/1) implies (x,y) C PA(~I). We are to show t h a t hi and h2 are bijections. Let us consider only the case of h2 since the case of hi can be proved similarly. Assume that a, b c A2 and a J: b. If h2(a) - h2(b), then h2(a) A~I ~;~(h~(b)) - 0 implies h2(a) A~ hi (~2(b)) - O' and so (h2(a),hl(q;2(b)) c P A ( ~ ) . Therefore (a, ~2(b)) c PA(~2) SO that hA1 ~2(b) - ~. Thus by (5) of Corollary 6.1.32, ~ ( b ) < ~2(a) ~ a _< b. In a similar manner 1 2 one can prove that b <_ a, that is, a - b. The contradiction obtained shows that 2 h~ is a bijection. It is likewise easy to see that the implication
(h2(z),hl(v)) ~ P A ( ~ ) ~
(Z,V) ~ PA(~2)
gives t h a t h l and hz are bijections. Hence it remains to show that
((hl(x),hl(y))
~
PA(~i) and (he(z),h2(v)) ~ PA(~Pl)
imply ( x , y ) e PA(~I) and (z,v) e PA(~Pz)) ~ hi and hz are bijections. We shall prove only the first case since the proof of the second one is similar. Let (h~(x),hl(y)) e P A ( ~ ) implies (x,y) e PA(~Pl). We are to show t h a t hi is a bijection. If a, b e A1, a ~ b and h~(a) - h~ (b), then
hi(a) Ai ~ i ( h l ( b ) ) - Q}t implies hi(a) A~ h l ( ~ l ( b ) ) - Q)t and so (hi(a), hi (~1 (b)) ~ PA (1/)~). Hence (a, ~1 (b)) E PA (~1) so that hA1 ~1 ( b ) - ~}. Therefore by (12) of Corollary 6.1.33, a <_ b. Similarly, one can prove that b <_ a, 1 1 t h a t is, a - b. The obtained contradiction shows that hi is a bijection. By a similar reasoning one can prove that h~ is a bijection.
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276
VI. Generalized Boolean Algebra and Related Problems
is a GI of A.
Now let h" A ~ A' and I' - (I{, I;) be a GI of A'. G
(1) Suppose that x
hi(x) Vri hi(y)
9h l l ( I ~ ) U h~l(I~) and y
9h~-l(I~). If x
9h~-l(I~), then
9I~ and h ~ l ( h i ( x ) Vti hi(y)) - h ~ l ( h i ( x Vi y)) C hi-l(I~).
Therefore x Vi y 9h ~ l ( h i ( x Vi y)) c hi-l(I~). If x 9h~-l(Ij), then hj(x)V~ hi(y) 9I~ and, therefore,
h ~ l ( h j ( x ) Vti hi(y)) - h ~ l ( h i ( x Vi y)) c hi-l(I~). Thus x Vi y (2) If x
9h ~ l ( h i ( x Vi y)) c hi-l(I~). 9 h~-l(I~), y 9Ai and y <_ x, then hi(x)
hi(y) <_' hi(x) so that hi(y) i
y If x
9hi-l(/~), y
9h ; l ( h j ( y ) )
9 A~and
9I~ C I~ U I~. Hence
9Aj and y 4 x, then y Aj ~i(x)
Since hi(x)
Let h
(~. Therefore
9 I~,we obtain hj(y)
C hjl(z~) c hll(/i)U
C o r o l l a r y 6.4.10. I - (Ii,h)
9 I~,hi(y)
9h~l(hi(y)) C hi-l(I~) C h l l ( I ~ ) U h2-1(Is
so that h j ( y ) 4 ' h i ( x ) . y
i
9Ij c I~ U g and thus
h21(/;).
9 A ~ A ~be a G.isomorphism of GBA's. G
[-1
Then
is a GI of A if and only i f h ( I ) - (hl(I1),h2(I2)) is a GI of A'.
C o r o l l a r y 6.4.11. The image of a GF under a G.isomorhism is a G F and the inverse image of a GF under a G.homomorphism is a GF. Thus, if h : A ~ A' is a G.isornorphism, then F = (F1, F2) is a GF of A if G
and only if h(F) = (h~(F1),h2(F2)) is a GF of A'. Proof. The proof is an immediate consequence of the G.duality. 6.5.
(i,j)-Atoms
D
and Pairwise Atomic Generalized Boolean Algebras
Since every pair from PA(~I)[-J PA(~2) is supposed to play the role of zero element of a GBA A, we can extend the notion of an atom to the case of a GBA, that is, to say, we can define the notion of an atom not as an element, but as a pair of elements of a GBA. Moreover, we shall establish a connection between an atom, defined as an element and an atom, defined as a pair. D e f i n i t i o n 6.5.1. An ( i , j ) - a t o m of a GBA A - {A1, A1, Vl,q01, i~, 4 ,e, A2, A2, V2, F2} is a pair ( a , a Vj 0 ) E Ai x Aj, where a r 0 and if (c,d) E Ai x Aj, c _< a, d _< a Vj 0 , then (c, d) - (a, a Vj 0 ) or (c, d) 9PA (~i). i j It is clear that a 9Ai \ {0} ~ (a,a Vj O)-~PA(~i), and if a 9A1, then (a,a V2 O) is a (1, 2)-atom ~ (a V2 O,a) is a (2, 1)-atom.
6.5. (i, j)-Atoms and Pairwise Atomic Generalized Boolean Algebras
277
P r o p o s i t i o n 6.5.2. Let A { A 1 , A 1 , V 1 , ~ 1 , ( 9 , ~ ,e, A2, A2, V2,~2} be a GBA. Then an element a E Ai is an atom of the BA Ai - {Ai, Ai, Vi, !)i, (3, <, e} i
if and only if the pair(a, aVj(3) is an (i,j)-atom of the GBA A (3, 4 , e, A2, A~, V~, q~}.
{A1, A1, V l , ~ l ,
Pro@ Let a c Ai be an atom o f t h e BA Ai. Then by (11) of Corollary 6.1.33, aVj(~ is an atom of the BA Aj. Assume that (c, d) E Ai x Aj is a pair such that c < a and d < a Vj @. Then it is clear that (c, d) - (a, a Vj (9) or (c, d) - ((9, d) c PA (~i) J
or (c,d) - ( c , @) 6 PA(P~) or (c, d) - ((9, (9) 6 PA ( ~ ) . Conversely, let (a,a Vj (9) be an ( i , j ) - a t o m of the GBA A. If a is not an atom of the BA Ai, then there exists an element c E Ai \ {(9} such that c < a. Then the pair (c, c Vj (9) satisfies the conditions:
(c, c Vj (9)-~ PA (p~).
c < a, c Vj (9 < a Vj (9 and i j D
P r o p o s i t i o n 6.5.3. Let c a t - {A1,A1,V1,~l,(9,~ ,e, A2, A2, Vg, p2} be a GBA and (a, a V 9 (3) ~ A~ • Aj. Then (a, a Vj (3) is an (i, j)-atom if and only if for every pair (c, d) E Ai x Aj, we have
a <_ c, a Vj (9 <_ d or (a Ai c, (a Vj (9) Aj d) C PA(9~i)-
i
j
Proof. First, we assume that (a, a Vj (9) is an ( i , j ) - a t o m and (c, d) E Ai x Aj is any pair. Then for the pair (a Ai c, (a Vj (9) Aj d) it is obvious that a Ai c < a, (a Vj (9) Aj d <_ a Vj (9 i
j
and hence
a Ai c -- a, (a Vj (9) Aj d - a Vj (9, that is a
aVj@
or (a Ai c, (a Vj (9) Ajd) 6PA(qDi)
since (a, a Vj (9) is an (i, j)-atom. Conversely, suppose that (a, a Vj O) E A~ x Aj and a pair (c, d) E A~ x Aj satisfies the conditions: c _< a, d <_ a Vj (9. If a <_ c and a Vj O < d, then a - c, i
j
i
j
a V j O - d. If (a Ai c, (a Vj @) Aj d) e PA(~i), then (c,d) c PA(~i) and thus (a, a My (~1) is an (i, j)-atom. D T h e o r e m 6 . 5 . 4 . For a G B A , 4 - {A1, A1, VI, ~I, @, 4 , e, A2, A2, V2, qp2} and an element a E A1, we have the equivalences"
ai(L)--(Ii--{xEAl <
" x<_a},I2-{yEA2 1
;" ( ~ l ( a ) V 1 (9,991(a))
z,: > ~ I ( a ) F ( R ) - ( F I - { x f f A is a G. ultrafilter.
" y 6 a}) is a prime GI,'
is a (1, 2)-atom
1 9~ l ( a ) ~ x } , f 2 - { y E A 2 "
<
',
~l(a)<_y}) 2
>
278
VI. Generalized Boolean Algebra and Related Problems
Pro@ By (3) of Corollary 6.3.4, it suffices to prove only the first equivalence. First, we assume that ai(L) is a left principal GI and (qDl(a) V1 QI, qo1(a)) is a (1, 2)-atom. We have to show that ai(r) is a prime GI, that is, at(c) is a maximal GI so that aI(L) is proper and has no property to be contained in a proper GI of .4. If aI(L) is nonproper, then a - e so that (~l(a) V1 l~, qPl(a)) -- ((~, Q)), which is impossible. Now we assume that there exists a proper GI I' - (I[,I~) such that de(L) < I ~. Then there exists an element b c I[ \ I1, and, therefore, b g I 1 , that is, b _< a is false. But in that case b _< ~)1 (a) and b :/= 2/)1(a) since otherwise 1
1
a, @l(a) E I [ , and so I' - (I~, I;) is nonproper. Thus b < ~)l(a). Consider the pair 1
(b, b V2 (9). It is clear that b < ~1 (a) V 1 1~, bve ~ < ~1 (a) and (b, b V2 O) g PA (~1), 1
2
which contradicts the fact that (~l(a) V1 i~, ~ l ( a ) ) is a (1,2)-atom. Hence aI(L) is a maximal, that is, a prime GI. Conversely, assume that de(L) is a prime GI and (qpl(a) V1 Q), ~ l ( a ) ) is not a (1, 2)-atom. Then there exists a pair (c,d) ~ A1 x A2 such that c < qPl(a)V1 (~), d ~ ~ l ( a ) , 1 2
(c,d) 7~ ( ~ l ( a ) V 1 (~), qDl(a))
and (c, d) c E PA (~1)- Clearly, C < ~ l ( a ) V1 O, 1
d < qpl(a) a n d 2
(c,d) # ( ~ l ( a ) V 1 0 , ~ 9 1 ( a ) )
give C ~ @l(a) V1 O, d < @l(a) or c < @l(a) V l O, d ~ ~ l ( a ) 1
1
2
or both c < ~ l ( a ) V 1 (~) and d < ~l(a). 1
2
Without loss of generality consider the case c <_ ~ l ( a ) V 1 (~, d <2 p l ( a ) . d < ~l(a) ~ 2
a < ~2(d) and so p 2 ( d ) g I 1 . 1
1
Then
On the other hand, if d E I2, then
d 4 a, that is, d < a V2 (9 - ~(~1(a) so that 2
(~92(~1 (a)) < ~2(d) ~ 1
~)l(a) <_ qD2(d). 1
Hence we conclude that a < p2(d) and ~)l(a) <1 p2(d), that is a V1 ~ l ( a ) -- e -- ~2(d) 1
and hence d - O. But this conclusion contradicts the fact that (c,d)-~ P A ( ~ l ) Thus d g I2 and, therefore, aI(L) is not a prime GI. D T h e o r e m 6.5.5. Let A - { A 1 , A 1 , V l , ~ l , O , 4 ,e, A2,A2, V2,~2} be a GBA and (a, a V j O ) E A~ x Aj. Then (a, a V j O ) is an (i,j)-atom ~ aGF F = (F1,F2), generated by the pair ({a}, {a Vj 0}), is a G.ultrafilter ~ a GI I (~2(F2), pl(F1)), generated by the pair (~i(a) Vi O, ~i(a)), is prime.
Pro@ By the G.duality and (3) of Corollary 6.3.4, it suffices to prove only the first equivalence. Assume that (a, a Vj (9) c Ai x Aj is an ( i , j ) - a t o m and F - ( F 1 , F 2 ) , where F~ - {x ~ A~ 9 a<_ x}, Fj - {y ~ Aj 9 aVj (~ <_ y} is a GF, generated by the i j
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280
VI. Generalized Boolean Algebra and Related Problems
Thus, as in the first case, a Vj (~ - (~ and so x c Fj or x ~ c Fi (or both x E Fj and x' c Fi). Furthermore, we are to prove that the GF (F1, F2) is proper. Indeed, if there exists an element x c A~ such that x c F~ and ~ ( z ) c Fj, then simultaneously, a _< x and a 4 ~ ( x ) so that a - O, which is impossible since (a, a Vj O) is an i
(i, j)-atom. To prove the converse, assume that F - (F1, F2) is the G.ultrafilter, generated by the pair ({a}, {a Vj 0}) so that Fi - {x c Ai" a <_ x} and Fj - {y c Aj " a Vj I~ ~ y}. i
2
Let us prove that (a, a Vj (~) is an (i,j)-atom. If (c, d) c A~ x Aj is any pair, then c c Fi or ~i(c) E Fj(~=~ r c Fi), but not both since F - (F1,F2) is a G.ultrafilter. Similarly, d E Fj or ~j(d) E Fi(e==~ r E Fj), but not both. Hence there arise the following cases" (1) c E F ~ a n d d E F j . Thena_
i
d) - (a, e) e PA
(a
(3) a Vj 0 <_ p~(c), a Vj 0 _< d, and, therefore, J J
(a
(a Vj e)
d) - (e, a
e)
9
Thus the sufficient condition of Proposition 6.5.3 holds and so (e, a Vj ~) is an (i, j)-atom. D D e f i n i t i o n 6.5.6. A GBA A - {A1, A1, V1, ~91, I~, 4 , e, A2, A2, V2, ~2} is said to be (i,j)-atomic if each pair (c, d ) C P A ( ~ i ) contains an ( i , j ) - a t o m so that for each pair (c, d) g PA ( ~ ) , there exists an (i, j ) - a t o m (a, a Vj (~) such that a <_ c and i
aVj(~<_d. J It is clear that for a GBA A, we have: A is p-atomic
~
A is (1,2)-atomic
~
A is (2, 1)-atomic.
Also, a GBA A - {A1, A1, V1, ~1, t~, 4 , e, A2, A2, V2, ~ 2 } is p-atomic ~ BA's Ai - {Ai, Ai, Vl, ~1, O, <_ e} are atomic. i
the
Indeed, if A is p-atomic and a E A~ \ {0} is any element, then the pair (a, a Vj (~) g PA ( ~ ) and, therefore, there exists an (i, j ) - a t o m (c, c Vj (~) such that c _< a and c Vj ~ _< a Vj (~. By Proposition 6.5.2, c is an atom in the usual sense i j and thus Ai are atomic. Conversely, let A~ be atomic and (a,b) ~ (Ai x A j ) \ P A ( ~ ) S O that a r (~ r b and b <_ ~i(a) is false. Then b Ai a r ~ and since Ai are atomic, there exists J an a t o m c c A~ such that c_< b A i a . T h e r e f o r e c 4 b, t h a t i s , c V j ( ~ <_ b a n d i j
6.5. (i, j ) - A t o m s and Pairwise Atomic Generalized Boolean Algebras
281
c <_ a. Again applying Proposition 6.5.2, we conclude that (c, cVj@) is the required i
(i, j)-atom. T h e o r e m 6.5.7. A GBA A {A1,AI,VI,~I,O,~ ,e, A2, A2, V2,p2} is p-atomic if and only if( Vi at, Vj bt) E Pv(~i), where {(at, bt)}tET is a family of tET1
tET2
all ( i, j ) -atoms of .4 and T1 U T 2 - T. Proof. First, assume that A is p-atomic and ( Vi at, Vj bt) -g Pv(~i). If V, at - a, t6T1
t6T2
tET1
Vj b t - b, then by ( 6 ) o f Corollary 6.1.32, (Fj(b), F i ( a ) ) C P A ( P i ) and, therefore, t6T2
there exists an ( i , j ) - a t o m (at,,, at,, Vj O) such that at,, < pj(b), at,, Vj 0 <_ r i j since A is (i, j)-atomic. Clearly, to c T - T1 U T2 implies to c T1 or to c T2. Without loss of generality assume that to c T1. Then at(, <_ a. On the other hand, ato Vj 0 <_ q~i(a) <_ ~i(ato) i j j and so (ato, ato Vj @) c PA (P~). This result contradicts the fact that (ato, ato Vj 0 ) is an (i, j)-atom. To prove the converse, let ( Vi at, Vj bt) c Pv(Pi), where T1 U T2 - T. We t6T1
t6T2
shall prove that A is p-atomic. Let us assume the contrary, that is, there exists a pair (c, d ) E PA(P~) such that for each ( i , j ) - a t o m (at, at Vj @), we have: at <_ c is i false or at V j @ <_ d is false. Let J
T1-
{t c T " at _< c is false} and T2 - {t c T " at Vj O <_ d is false}. i j
It is obvious that T1 U T 2 - T. Moreover, by Definition 6.5.1, we have
(at Ai c, at Vj O) 6 PA(~/) for each t C T1 and
(at, (at Vj (9) Aj d) c PA (~i) for each t C T2. Therefore
at Vj 0 ~_ ~i(at) Vj ~i(c) for each t c T1 J and
at <_ ~j(at Vj (9) V~ pj(d) for each t c T2. i
Hence
at Vj 0 _~ qpi(c) for each t c T1 and at <_ pj(d) for each t c T2 j i since
(at Vj O) Aj pi(at) -- O and at Ai ~j(at Vj 0 ) -- 0 for each t c T. Clearly,
V ~ at <_ ~j(d), i
t6T2
~ t6T1
(at Vj @) _< ~i(c) j
282
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
and if V~ at - a, Vj bt - b, t h e n tET2
tET1
a <_ ~ j ( d ) ,
b <_ ~ ( c ) j
and (a, b)
9Pv(p~)
since T2 t2 T1 - T. Therefore
~ i ( ~ j ( d ) ) <_ ~i(a) <_ b <_ ~i(c), J
J
t h a t is d <_ ~i(c)
J
J
and thus (~j(d), ~i(c)) 9P v ( ~ i ) . But in t h a t case, by (6) of Corollary 6.1.32, we have (c, d) 9P~ ( ~ ) , which is impossible. Hence ,4 is p-atomic.
6.6. G e n e r a l i z e d B o o l e a n Factor A l g e b r a s We shall define a factor algebra of a G B A A = {A1, A1, V1, ~1, ~=), ~ , e, A2, A2, V2, p2}. Let I = (I1,/2) be any proper GI of A and F = ( ~ 2 ( 2 2 ) , ~ 1 ( / 1 ) ) b e the corresponding GF. If E~ = A~ U / j t2 ~i(I~), t h e n Ai • / j = { 0 } , Ai • ~i(I~) = {e} since A1 V1A2 = {(~, e}, and Ij A ~i(Ii) = f~J since I = (I1,/2) is a proper a I .
P r o p o s i t i o n 6 . 6 . 1 . Let A = {A1, A1, V1, ~1, (~, 4 , e, A2, A2, V2, ~2} be a G B A and ~ be binary relation on the set Ei, defined in the m a n n e r as follows: i
(1) I f x, y C Ai, then x~y
,z----5, ( x - y ,
i
.,
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ja
y ) v~ e
y-xEZj)
; :,
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= y - x e I~). i
j~
= x - y, i
(2) / f x c A~, y c Ij U ~ i ( I i ) , then i ~:
i
iG
:. ( ( ~ - y ) v j e) = ~ - y e I j , ic;
(3) I f x, y
J
JG ( y - ~ ) v~ e Jc;
:
y - x e I~). i
9Ij U ~i(Ii), then x~y
i
~.
( x - y, y - x c I~) ~: :. i(;
~: :~ ( ( x - y) v j e = it;
iG
9 - y, ( y - x ) v~ e = y J
iC
x
9I j ) .
J
Then ~ is an equivalence relation on Ei such that elements f r o m Ij and y)i(Ii) i
belong to different equivalence classes. Moreover, if x, y 9Ij or x, y 9~i(Ii), then x ~ y ,z---5, x ~ y. i j Proof. First, let us prove t h a t ~ is an equivalence relation on Ei. It is obvious i
t h a t x x, x for each x i
9E~ since 11 N / 2 = {(~}. Also, by (2), the implication
x ~ y ---5, y ~ x is clear for each x, y i
i
9Ei. Therefore it remains to show x ~ y i
and y ~ z imply x ~ z. We shall consider the following cases: i
x, y, z
9Ai;
x
9A~, y, z
9Ij U ~i(Ii);
x, y
9Ai,
z
9Ij [2 ~i(I~)
6.6. G e n e r a l i z e d B o o l e a n F a c t o r A l g e b r a s
283
and
x, z ~ A~,
y ~ Ij U ~ i ( I i ) .
T h e r e s t r i c t i o n ~ IA, is an e q u i v a l e n c e r e l a t i o n on Ai since Ii is an ideal in t h e usual sense. H e n c e t h e case x, y, z ~ Ai is obvious. Since all t h e o t h e r cases are p r o v e d similarly, we shall consider only t h e case w h e r e x ~ A~ a n d y, z ~ I j U ~ ( I ~ ) . B y (2) a n d (3), we o b t a i n
x,~y i
,,t---5, (x - y ~ Ii, y - x ~ I j ) i(;
j(;
and y~
i
z ,,+---5, ( y - z , i(;
z - y c Ii) i(;
so t h a t
a~ ~j(y) c I~, y Aj ~(~) c Ij, y a~ ~j(z), ~ a~ ~j(y) e I~. T h e r e f o r e G D L 1 , G D L 2 a n d Definition 6.2.1 i m p l y
(x A~ ~j(y)) v~ (y A~ ~j(~))- ((~ A~ ~j(y)) vj y) A~ ((x A~ ~j(y)) v~ ~j(z))(~ v~ y) A~ (~j(y) vj y) A~ (~ v~ ~j(~)) A~ (~j(y) v~ ~j(~)) = (x vj y) A~ (~ v~ ~(~)) A~ ( ~ ( y ) v~ ~j(~)) 9I~ since I = ( I 1 , / 2 ) is a GI. Clearly, i
Following (2) of Definition 6.2.1, x Ai ~ j ( z ) = x - z e Ii. i(;
In a similar m a n n e r one can show t h a t z - y E Ii a n d y i (;
J c;
x E Ij i m p l y
Z--XCIj.
H e n c e x ~ z and, therefore, ~ is an e q u i v a l e n c e r e l a t i o n on Ei. i
i
F u r t h e r m o r e , let us prove t h a t e l e m e n t s from Ij a n d ~ i ( I i ) b e l o n g to different e q u i v a l e n c e classes. If x c Ij, y c y:i(I~) a n d x ~ y, t h e n t h e r e is an e l e m e n t z E I~ i
such t h a t y = y)i(z) as y c qpi(Ii). Moreover, x ~ y implies x Ai ~ i ( Y ) , Y Ai ~ j ( x ) C Ii, i t h a t is, a n d so z Vj x c ~ ( I ~ ) . O n t h e o t h e r h a n d , z ~ I~, x ~ I o i m p l y z Vj x ~ Ij a n d h e n c e z Vj x ~ Ij N ~ i ( I i ) , which is impossible since I 3 A p i ( I i ) = ~ . Finally, let x, y ~ I o or x, y ~ ~ ( L ) . It is obvious t h a t Ij U ~ ( I ~ ) C E~ C~A j . T h e r e f o r e , on t h e one h a n d , by (3), for a pair x, y ~ Ij U ~ i ( I i ) c Ei, we have x ~ y ~ (x - y, y - x E I~), and, on t h e o t h e r h a n d , by (1), for t h e s a m e pair i
i(,
i(;
x, y ~ Ij U ~ i ( I i ) C Ei, we h a v e x ~ y .z----5. (x - y, y - x ~ I~). T h u s , if x, y ~ Ij j
or x, y C ~ ( I ~ ) , t h e n x ~ y i
i(;
i(;
.z--5. x ~ y. j
Let us d e n o t e t h e set of all ~ - e q u i v a l e n c e classes by E i / I ,
E{/I - {[x] {- x s E{}.
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VI. Generalized Boolean Algebra and Related P r o b l e m s
Thus the algebra . A / I - {El/I, [-]1, l-]1, (1)1, [l~], ~ F, [e], E2/I, R2,112, ~2} is a G B A t h a t we call a G.Boolean factor algebra of the G B A A - {A1,A2, Vl, ~1, (9, 4 , e, A2, A2, V2, ~2}. Moreover, if we consider the pair n - (nl, n2), where the maps ni" Ai ~ E i / I are defined as ni(x) - [x] i, then it is easy to see t h a t n ( r t l , rt2) " fl[ ---+ .A/I. This G . h o m o m o r p h i s m is called a natural G.homomorphism. G
Finally, let us consider any G . h o m o m o r p h i s m h - (hi, h2)
.4 9 ~ A ~. Then, by G
Example 6.4.2, h - 1 ( O ') - (h[l((~'),h~l((~')) is a proper GI of .4. Let A / h - I ( O ') be the G.Boolean factor algebra. If g-(g1, g2), where gi" Ei/h-l((~') ~ A~ are de9 ~a .4 ~ is a G.isomorphism. fined as gi([x] i) - hi(x) , then g - (91,92) A/h-l(~)') Indeed, by (7) of T h e o r e m 6.4.7, it suffices to prove only t h a t gl([X] 1) - (9' and g2([y] 2) - (9' imply [x] 1 - [(9] and [y]2 _ [(9]. But this implication is an immediate consequence of the definition of g - ( g l , 9 2 ) . 6.7. Generalized Fields of Sets and the Generalized Field Representation of a Generalized Boolean Algebra D e f i n i t i o n 6.7.1. Let X be a n o n e m p t y set, Ai(X) be a ring of subsets of the set X , t h a t is, Ai(X) be closed under the finite set-theoretic operations intersection and union, ==<: be a quasi order relation on A I ( X ) u A2(X) such t h a t ==~ IA,(x) = C , A I ( X ) N A 2 ( X ) = {2~, X } and A c Ai(X) implies X \ A c Aj(X). If there exists maps []i: Ai(X) ~ A j ( X ) satisfying the conditions:
(1) A ~ [A]~, [ [A]~]j = A and X \ [A]~ = IX \ A]j for each set A e A~(X). (2) [ ] i o n = No([ ]i, []i) a n d [ ] i o U = Uo([ ]i, []i) s o t h a t [ANB]i = [AIiA[B]i and [A U B]i = [A]i U [B]/ for each pair (A, B) c A i ( X ) x A i ( X ) , then we shall say t h a t Jc(X) = {AI(X)UA2(X),N, u, \, ==<:, []1, []2]} is a G.field of sets (briefly, GFS) for which X is the basic set. Clearly, []1 and []2 are bijections and []i - - [ ] ; 1 It is easy to verify t h a t the equivalence (A =<: [A]i and [ [A]i]j - A) ,z---->,A ==~ [A]i ==<: A holds for each set A E Ai(X). Indeed, A :=<: [A]i and [[A]i]j - A imply [A]i =:<: [[A]i]j - A. On the other hand, if A ==<: [A]i =<: A, then A =:< [A]~ ==< [ [A]/]j =:< [A]~ =:< A and so A ~ [[A]~]j ==<: A. Therefore A c_ [[A]~]j c_ A since ==<: latex) and thus [ [AJ~]j - A. Obviously [2~]i- o and [ X ] i - X. Moreover, for each set A e Ai(X), the set [A]i is a smallest set from A j ( X ) (with respect to the set-theoretic operation inclusion) such t h a t A ==~ [A]~. Indeed, if B c ,Aj(X) is any set such t h a t A ==~ B C [A]i, then [A]i ==<: A ==<: B c [A]i, t h a t is [A]i c_ B c_ [A]~ and thus B -
[A]~.
6.7. G e n e r a l i z e d F i e l d s of Sets a n d . . .
289
For a GFS ~'(X) - { A I ( X ) U . A 2 ( X ) , N , U, \, ==<, []1, []2} let us define binary operations ni, Ui " (A~(X) x A i ( X ) ) u ( A j ( X ) x A i ( x ) ) --+ A i ( X ) as follows: AN~B-ANB,
AU~B-AUB,
if ( A , B ) r ( A i ( X ) • A i ( X ) ) \ {(A, [X \ A]j) " A r A~(X)}, and A Ni [X \ A]j - ~, AN~ B - [A]j N~ B,
A Ui [X \ A]j - X if A c A ~ ( X ) ;
A U~ B - [A]j U~ B if ( A , B ) e A j ( X ) x A~(X)
so that ANiB--
~'
if A - X \ B , if A ~ X \ B ,
( [A]jNB,
A U i B - ~X'
if A - X \ B , if A C X \ B .
[ [A]jUB,
Hence for ( A , B ) c A j ( X ) x A~(X), we have A n~ B C_ B,
A o~ B c_ [A]j ~
A and B c_ A U~ B,
A ~
[A]j c_ A U~ B.
From the above arguments, we immediately conclude that [A n~ Bli - [A]i nj [B]i,
[A Ui B]~ - [ A ] i
Uj
[B]i if (A, B) r A i ( X ) x A i ( X )
and [A N, B]~ - [ IA]j N~ B]i - A Nj [B]~, [ A U ~ B ] i - [ [ A ] j U ~ B ] i - A U j I B I i if (A,B) E A j ( X ) x A i ( X ) . Hence it is not difficult to see that every GFS can be treated as a GBA A(~'(X)) with ~ instead of 4 , c instead of <, ~ and X instead of 0 and e, i
respectively, the set-theoretic complementation \ instead of Wi, []~ instead of X , Oi and U~ instead of Ai and Vi, respectively. (The converse of this statement is Theorem 6.7.5.) Indeed, let us first consider the question of coordination of the quasi order - - < and the lattice operations ni, Ui to show that A ==< B .e---->.A Ui B - B for each pair ( A , B ) c ( A i ( X ) x A i ( X ) ) u ( A j ( X ) x A~(X)). If (A, B) c A ~ ( X ) ( x A ~ ( X ) , then A=:< B .<-->. A C_ B ..e->. B - A U B ..~->. B - A U~ B, and if ( A , B ) c A j ( X ) x A i ( X ) , then A ::::< B ~
[A]j c_ B ..v-->. B - [A]j u B ~
B - [A]j u~ B ~
B - A u~ B.
Now the examination of axioms L1, L2, GL3, GL4, GDL1 and GDL2 reduces to simple calculations which readily show that A(.T'(X)) - {AI(X),N1, Ul, \ , ~ , --<:, X, A i ( X ) , N2, U2, \} is a GBA. A GFS 5r(X) - {.41(X) U A2(X), N, U, \, ==<, []1, []2} is said to be reduced if for every pair of distinct points x, y c X there exists a set A c A i ( X ) such that x E A and y - c A .
290
VI. Generalized Boolean Algebra and Related Problems
D e f i n i t i o n 6.7.2. A G.field representation of a GBA .4 - {A1, A1, V l , 991, (~), 4 , e, A2, As, V2,992} is a pair ( f , X ) , where X is the basic set of a GFS Jc(X) considered as a GBA A(Jz(X)) = {A1(X),cll,U1, \,;g, ===~,X,A~(X), Ns, Us, \}, and f 9 A ~ A ( ~ ( X ) ) is a G.isomorphism such that f - (fl, f2), G
where f ~ ' A ~ ~ A~(X) are bijections, and the following conditions hold: (1)
s o Ai -- rli o ( f i , s
and s o Vi -
Ui o ( f i , f i )
so that
f~(~ A~ y) - f~(x) n~ s
~nd f~(~ v~ y) - s
u~ f~(y)
for each pair (x, y) c Ai x Ai. (2)
f i o Ai -- rli o
(fj, fi)
a n d f~ o Vi - Ui o
(fj, fi)
so that f~(x/~
y) - f j ( x ) n ~
s
y)
f~(y) - [ f j ( x ) ] j
n~ f~(y)
and v~
=
f~(x) u~ s
=
[f;(x)]j u~ f~(y)
for each pair (x, y) E Aj x Ai. (3) \ o fi - fj o 99~ so t h a t X \ fi(x) = fj(99i(x)) for each x 6 A~. Note t h a t here we also identify these (reduced) G.field representations (f, X) and ( I ' , X ' ) of a GBA ,4 = {A1,A1,V1,991, O , ~ ,e, As, As, V2,992} which are equivalent in the sense as follows: there exists a bijection h : X ~ X t and a G.isomorphism h* = (h~,h~) : A ( 5 ( X ) ) ~ A ( ~ ( X ' ) ) such that h~(A) = {h(x) : x c A} for every set A c X i ( X ) . L e m m a 6.7.3. If (f , X ) is a G.field representation of a GBA .4 = {A1, A1, V1, 991,O, ~ ,e, A2, As, V2,992}, then for each x c Aj we have [fj(x)]j = f~(x V~ 0).
Proof. By (6) of Theorem 6.1.16, x c Aj implies x V~ 0 6 x and, therefore, by (1) of Definition 6.7.1 and the G.isotonicity of f = (fl, f2), we have f i ( x Vi (9)==3
[fj(x)]j,
that is fi(x Vi (~) c_
[fj(x)]j.
On the other hand, the same arguments give fj(x)==~ fi(x Vi (9) and, hence, by the definition of the set [fj(x)]j, we have fj(x)==< [fj(x)]j c_ fi(x Vi (9). Thus
[fj(x)]j - f i ( x Vi 0).
[3
We are now ready for the principal result of this section; namely, the next theorem characterizes all the reduced G.field representations of a GBA .4. The existence of a reduced G.field representation will be a consequence of this theorem and of the results of previous sections. T h e o r e m 6.7.4. There exists a one-to-one correspondence between the reduced G.field representations of a GBA A and the Stone families of prime GI's of X.
Pro@ Let S be a Stone family of prime GI's of a GBA A. The reduced G.field representation (f, X) of A can be associated with S in the manner as follows: let
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VI. Generalized Boolean Algebra and Related P r o b l e m s
Conversely, if x1 ~ f j ( a ) A~ f~(b) - [fj(a)]j ~ k ( b ) , then
f j ( a ) Ni fi(b) - [fj(a)]j N f~(b) so t h a t x1 ~ [fj(a)]j ~ f~(b) a n d s o x i ~ f ~ ( a V ~ ) ~ f ~ ( b ) . Therefore a V i @ g I ~ z--->, a - g I j and bgI~. Now, Definition 6.2.9 gives aA~b-gIi so t h a t x i e f~(aA~b) and thus f~(aA~b) - f j ( a ) ~ f ~ ( b ) . Furthermore, if x~ ~ f~(a V~ b), where a ~ A j , b ~ Ai, then a V~ b--gI~. By (1) of Definition 6.2.1 the case a ~ Ij, b ~ I~ is impossible. Hence a-g Ij or b g I~ (or both a g Ij and b g/~). We shall consider only the cases a g Ij, b ~ /~ or a g Ij,
bg~. If a-g I j, b ~ I~ and b -
~ j ( a ) , then
f j ( a ) ~ fi(b) - [fj(a)]j ~2~ f~(b) - [fj(a)]j t3~ f i ( ~ j ( a ) ) - [fj(a)]j ~ ( X \ f j ( a ) ) - X so t h a t x i ~ f j ( a ) ~ fi(b). Now we assume t h a t b --/: ~j(a). Then
f j ( a ) tAi fi(b) - [fj(a)]j Ui fi(b) - [fj(a)]j ~ fi(b) and, therefore, g-EIj ~
g V i {~--EI i ,g------5,xI E f i ( g V{ ~ ) -- [fj(g)]j,
t h a t is, x1 ~ f j(a) Ui k ( b ) . I f a - g I j , b-gIi, then, by (2)of Theorem 6.2.10, the case b - ~ j ( a ) is impossible. Hence f j ( a ) U i fi(b) - [fj(a)]j U fi(b) and a - g I j , b-gIi imply a V~ O - g I i , b-gIi so that x1 e k ( a Vi O ) U fi(b) - [fj(a)]j U fi(b) - f j ( a ) Ui fi(b). Conversely, let x , c f j ( a ) Ui fi(b) and b - ~j(a). Then
f j ( a ) t& fi(b) - [fj(a)]j tAi f i ( ~ j ( a ) ) - [fj(a)]j tAi ( X \ f j ( a ) ) - X and, by (2) of Theorem 6.2.10, a-EIj or b - qpj(a)-EIi. It is clear t h a t always a V~ b-~Ii so t h a t x1 c f i ( a Vi b). Now let b 7~ ~j(a). Then f j ( a ) Ui f~(b) [fj(a)]jU fi(b) and so x1 E [fj(a)]j - fi(aVi(~) or x i c fi(b) (or both x i c [fj(a)]j and x i E fi(b)). Clearly, in all cases a Vi b E I i so that x i c f i ( a Vi b) and thus f i ( a Vi b) - f j ( a ) tO~ fi(b). We have shown that the conditions (1)-(3) of Definition 6.7.2 are fulfilled and, therefore, f - (fl, f2) " A - {A1, A1, Vl, q;1, O, 4 , e, A2, A2, V2, P2} -+ f(.A) A ( . T ' ( X ) ) - { A I ( X ) - { f l ( a ) - { x i " a-gI1 ..r a v2 ~ ) g I 2 , I - (11,12) c S } " a c A 1 } , N 1 , U I , \ , , ~ , ===<,X,.A2(X) - {f2(b) - { x I " b-gI2 ~ bV10cIl, I(11,12) E S} 9 b EA2}, C12,tO2, \} is a G.homomorphism of the GBA A onto the GFS ~ ( X ) , considered as the GBA A ( . T ( X ) ) . To prove t h a t f is a G.isomorphism, it remains to show only t h a t fl and f2 are bijections so that by virtue of (7) of Theorem 6.4.7, it suffices to show t h a t if f l ( a ) - f2(b) - ~, then a - b - O. But this implication immediately follows from the definition of f~ and (1) of Theorem 6.2.16. Furthermore, let us prove t h a t ~ ( X ) is reduced. Indeed, if x i , xi, E X and xx ~ x i,, then by the definition of the set X, we have I - (I1, I2) r I ' - (I~, I~). Therefore since I and I ' are prime GI's, there exist elements a, a' E A1, b, b' E Az such t h a t a f t I1,
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a'-EI1,
a' E I~, b E I2, b g I ~ ,
b'gI2,
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VI. G e n e r a l i z e d Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
and thus x - ~ f j ( b ) U [fj(b)]j, that is, b E I x. Hence we conclude that Ix = (I~, I f ) is a GI. Moreover, the proof of (1) immediately implies that Ix is a proper GI. It remains to prove that Ix = (I~, I~) is a prime GI, that is, the condition of Definition 6.2.9 holds. First, assume that a, b ~ A~ and a A~ b E I x. Then
x-~ f~(a A~ b)U [f~(a A~ b)]~ - (f~(a) n f~(b)) U ([f~(a)]i A [f~(b)]~). If we assume the opposite, that is, a g I x, b g I x, then
x E (fi(a)U [f~(a)]~)N (f~(b)U [f~(b)]~).
(I)
Since a Ai b E I ,x, (I) immediately implies
x E f~(a)rq (X \ [f~(a)]~) rq (X \ f~(b)) rq [f~(b)]~
(II)
x E (X \ f~(a)) n [f~(a)]~ rq f~(b)0 (X \ [f~(b)]~).
(III)
or
By the definition of maps f~, in the case (II), we have
xE[fi(a)]~ - fj(a Vj e) ,z---->,a Vj e E I f ~
a c Ix ~
x E fi(a),
which is impossible. Similarly,
x--~ f~(b) ~
b E I~ ~
b Vj 0 E I f ~
x-E fj(b Vj 0) - [fi(b)li,
which is also impossible. Using similar arguments, one can prove that (III) is impossible. Thus if a, b E Ai and a A~ b E I~, then
x--~fi(a) U [fi(a)]i or x - c f i ( b ) U [f~(b)]i ( o r both x-E (Si(a)U [S~(a)]i) U (Si(b)U [S~(b)]~) ) and s o a c I x o r b E I x (or b o t h a c I x a n d b E I X ) . Finally, let a E Aj, b E Ai and a Ai b E I x. Then
x-E fi(a Ai b) U [fi(a Ai b)]i - (fj(a) ni fi(b)) U [fj(a) ni fi(b)]i = = ([fj(a)]j ni fi(b)) U ([fj(a)]j Nj [fi(b)]i) -
= ([fj(a)]j N fi(b)) U ( f j ( a ) N [fi(b)]i) and, using the arguments as above, we obtain a E I f or b E I x (or both a E I f and b E I / ) . Thus Ix - (I~, I f ) , where I.~ - {a c A~" x--~f~(a)U [fi(a)]~}, is a prime GI. Now let us prove that S - {Ix - (I~, I f ) " x E X } is a Stone family of prime GI's. Indeed, let a c Ai \ {fg} be any element. Then fi(a) ~ ;g and so, there exists an element x E fi(a). But by the definition of maps f~, we have a g I x and, therefore, (2) of Theorem 6.2.16 implies that S is a Stone family. Thus, we have constructed two maps Ct I and a2, where O~1 associates the reduced G.field representation of A with every Stone family S of prime GI's of .4 and vice versa: a2 associates the Stone family S of prime GI's of A with every reduced G.field representation (f, X) of ,4.
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VI. Generalized Boolean Algebra and Related Problems
(f', X'), then by the definition of (f', X') the points of X ' are in the one-to-one correspondence with the prime GI's of S and, accordingly, with the points of X. Moreover, following the corresponding definition,
f{(a) - { x e X " a-~I f ~ for each element a
a vj O-~I~, Ix - ( I { , I ~ )
e S}
9Ai and
e f'(a)
a-
Ix
so that f~(a) = fi(a) for every element a
a
v; e-cI
x c f
(a)
9Ai. This completes the proof.
D
By virtue of Corollary 6.2.14, the family S of all prime GI's of a GBA ,4 is a Stone family and on the strength of the preceding theorem, there exists a reduced G.field representation of.4 associated with S. Thus the following principal statement is valid. T h e o r e m 6.7.5. Every GBA is G.isomorphic to a reduced GFS.
Proof. Indeed, the family of all prime GI's of a GBA A is a Stone family. By Theorem 6.7.4, to this Stone family there corresponds the G.field representation ( f , X ) of A and from Definition 6.7.2, it follows that the GFS De(X), considered as the GBA A ( ~ ( X ) ) = {AI(X) = {fl(a) : 9A 1 } , n l , U l , \ , ~ , ==<,X, A 2 ( X ) = {f2(b) : b 9A2}, N2, u2, \}, is G.isomorphic to A. [~ C o r o l l a r y 6.7.6. For each real number m > 0 and every non-zero element a 9Ai there exists a G.quasi measure # = (p1,#2) on a GBA A = {A1, A1, V1, pl, O, ~ ,e, A2, A2, V2, ~2} such that #i(a) = # j ( a V j O) = m .
Pro@ Let m > 0 be any real number and a c A~ \ (2~} be an arbitrary element. Then, by definition, fi(a) -r ;g =/=f j ( a Vj O) and, therefore, there exists an element b 9f~(a). If we define a pair # = (#1,#2) as
{
?T~, if b e fi(x) if b-~ fi(x) for each x E Ai
-
0,
and
{77~, 'J(Y)
-
0,
if b Vj 0 c f j(y) if b Vj @-E f j (y) for each y c Aj '
then it is not difficult to see that # = (~tl, it2) is the G.quasi measure on ~4 such that #i(a) = # j ( a V j (~) -- m . D The reduced G.field representation of a GBA A, associated with the family of all prime GI's, is called the perfect G.field representation of .4. Theorem 6.7.5 concludes one part of the study of representations of a GBA. In the next section we shall investigate the second part of the modification of Stone's representation theorem.
6.8. Bitopological Representation ...
297
6.8. B i t o p o l o g i c a l R e p r e s e n t a t i o n of a G e n e r a l i z e d B o o l e a n A l g e b r a Let A = {A1,A1,VI,g~I,O, 4 , e , A2, A2, V2, q~2} b e a GBA, S = { I = (11,/2)} be any Stone family of prime GI's of A and (f, X) be the reduced G.field representation of A associated with S in the sense of the first part of Theorem 6.7.4. We can endow the set X with the topologies rl and 72 by taking the rings of sets .41(X) and A 2 ( X ) of the GFS 2r(X) = { A I ( X ) U A 2 ( X ) , N, U, \, ==<, []1, []2} as the corresponding bases of the open sets. Clearly, t h e / - o p e n sets have the base consisting of j-closed sets since A i ( X ) = co A j ( X ) and, therefore, (1) of Corollary 3.1.6 implies that p-ind X = 0. On the other hand, if we assume that x, y E X, x -r y, then/V(X) is reduced implies, in particular, that there exist A, B c A I ( X ) such that x c A, y g A and x g B, y c B. It is clear that
A, B E rl NCOr2, X \ A, X \ B E r2 DICOrl so that
U(x) = A c 7-1, U(y) = X \ A c T2, V(x) = X \ B C ~-2 and V(y) = B E T1 are disjoint neighborhoods of the points x, y E X and thus, by (9) of Definition 0.1.6, (X, T1, T2) is p-Hausdorff.
T h e o r e m 6.8.1. Let S = {I = (I1,/2)} be a Stone family of prime GI's of a GBA A and let (f, X ) be the corresponding reduced G.field representation of A. Then the BS (X, T1,7-2) is FHP-compact if and only if S contains all the prime GI's of .4, that is, if and only if (f, X ) is the perfect G.field representation of A. Proof. First, assume that S = {I = (I1,/2)} is the Stone family of all prime GI's of the GBA A. It suffices to prove that every p-open covering U C {{fl(a)
9 a cA1}, {f2(b) " b c A2}}
of X such that b/N T1 r ~ r ~ ~'17-2 has a finite subcovering. Assume the opposite, that is, there exists a p-open covering { { f l ( a n ) : rt C N } , {f2(bm) : rrt E M } } of X so that hEN
and
mEM
# ( U Sl/an/)o ( U hEN1
mEM1
for any finite subsets N1 c N and M1 c M. Therefore, alVla2V1...Vlan~e,
bl V2 b2 V2 "" V2 bm C e
and ~1(al V1 ae VI'-" V1 a~) ~ bl Ve be V e " . V2 bm is false for any finite subsets N1 = { 1 , 2 , . . . , n } C N, M1 = { 1 , 2 , . . . , m } C M. Indeed, if a l V1 a2 V1 ... V1 an ~: e, then f l (al V1 a2 V 1 . . . V1 an) 7s f(e) - X
and so
f l ( g l Vl g2 Vl "'" Vl
an)
--
f l ( g l ) Ul f l ( g 2 ) U l " ' " Ul/1(an)
-
-
298
VI. Generalized Boolean Algebra and Related Problems = fl(al) U fl(a2) U... U
since f -
(fl,f2)
al V l a 2 V 1 . . . V l a n
fl(an) 5r X,
is a G.isomorphism. The same is true for bl,b2,...,bm. If - - a , b l V2b2V2---V2bm - b and ~bl(a) 4 b, t h a t is, g91(a) < b, 2
then f - ( f l , f2) is a G.isomorphism implies t h a t q~l(a) _< b ,e---->,f2(~l(a)) C_ f2(b) ,e----->,X \ / l ( a ) c_ f2(b)<--->, fl(a) U f 2 ( b ) - X since 4
2 IA,--~
and =:< A , ( x ) - C _ .
Thus the simultaneous fulfilment of all three cases al Vl a2 V1 "'" V1 an r e, bl V2 b2 V2 "-. V2 bm r e and ~31(al V1 a2 V l " ' " Vl
an) ~ bl V2 b2 V 2 " "
V2 bm
is false enables us to conclude t h a t
X ~ (
U fl(an))U
nCN1
(
U
mCM1
f2(bm))
for any finite subsets N1 C N and M1 C M. Let I - (I1,/2) be the GI generated by the pair of sets (B1 - {an " n E N}, B2 - {bin 9 m eM}). Then I - (I1,12) is proper since the contrary means t h a t e c I1 N I2 and, by Theorem 6.2.5, there exists, in particular, a finite sequence of elements a l , a 2 , . . . ,ak E B1 such t h a t al V l a 2 V 1 - - . V l a k - e, which is impossible. Let us consider a family of all proper GI's ordered by the partial order < defined in the part preceding Proposition 6.2.4. Then, by virtue of the latter proposition, every chain in this family has the upper bound, also defined by this proposition. Therefore Zorn's lemma implies t h a t this family has a maximal element, namely, a prime GI I. - (I~,I~) such t h a t I < I.. Clearly, an c Ii ~ ~ an V2 (9 c I~ for each n E N and bm C I~ ~ bm V1 I~ E Ii ~ for each m c M. Also, I* c S. But ( f , X ) is the reduced G.field representation of the G B A A which corresponds to S and, by (.) in the proof of Theorem 6.7.4, xi. g f l ( a n ) and xI. gf2(bm) for each n c N and m c M. Thus xL g ( U fl(an)) U ( U f2(b~)) - x . This contradiction proves the first part
nCN
mCM
of the theorem. Conversely, let (X, rl, r2) be F H P - c o m p a c t and there exists a prime GI I (I1,/2) such t h a t I g S so t h a t I r Ix for each point x c X. Therefore for each point x E X, the following conditions hold: (1) There exists an element a E A1 such t h a t a c
I1 ,z----F,a V2 0 E / 2
and a-gI~ ,e----->,a V2 ( g g I ~ ,e---->,x c f l ( a )
and there exists an element b c A2 such t h a t b g / 2 <---> b V 1 O g I 1
and b E I~ ,e---->, b V 1 1~ E I~ ,e----->,x-gf2(b).
(2) There exists an element c E A1 such t h a t
c-CI1 ~
c V20g/2
and c c I~ <---> c V2 0 C I~ ,e---->x g fl(c)
6.8. Bitopological Representation ...
299
and there exists an element d c A2 such t h a t d c / 2 ,z---5, dV1 (3 c I1 and d - ~ I ~ ~ Since for each point x e also prime, in both cases (1) d as ~1 (c), that is, b = ~ l ( a ) , condition (1) above. Indeed, for each element that
X the GI Ix = ( I ~ , I ~ ) is prime and I = (I1,I2) is and (2) one can consider the element b as ~ l ( a ) and d = q~l(C). As will be seen below, it suffices to apply x c X, there exists, by (1), an element a x c A1 such
ax V2 O c I2 and ax ~ I~ ~
a~ E I1 ~
dV1 ( ~ g I ~ ,,+----5x c f2(d).
{fl (ax): x c X} is
1-op
ax V2 O ~ I~ <---->.x E fl (ax).
cov i g of X.
b c I2 \ { e }
element. Then f2(b) ~ ;g and it is obvious that {{fl(ax) : x E X}, {f2(b)}} is p-open covering of X so that, there exists its finite subcovering since (X, T1, ~-2) is FHP-compact. If this finite subcovering does not contain the set f2(b), t h a t is, if it has the form { f l ( a x k ) : xk c X, k - 1,n}, then f l ( a x l ) U f l ( a x 2 ) U . . . U f l ( a x , , ) X and so axl V1 ax~ V1 "'" V1 a~,, = a = e since, like in the proof of the first part of this theorem, for a :/: c, we obtain f l (a) ~ X~ that is, f l ( a X l V1 ax 2 V I . . . V1 ax,,) = f l ( a X l ) [-Jl f l ( a x 2 ) U l . . . [-Jl f l ( a z , , ) ~ X ,' :,
> fl(aXl)
;.
9f l ( a x 2 ) U " . U f l ( a x , , ) r X .
But ax~ c I1 for each k = 1, n and, therefore, by (1) of Definition 6.2.1, a = e c I1, which is impossible since I = ( / 1 , h ) is a prime GI. Thus f l ( a ) r X and hence finite subcovering has the form {{fl(ax~) : xk c X, k = 1,n}, {f2(b)}}. Thus f l ( a x l ) U f l ( a x 2 ) U . . . U fl(ax,~) U f2(b) - X ,', > z, ,, f l ( a ) U f2(b) = X z----->, ~ l ( a ) < b, 2
which is also impossible since a c I1, b c / 2 and I = (I1,/2) is a prime, t h a t is, a proper GI. Therefore I c S so t h a t S contains all prime GI's of the GBA A. D By virtue of the reasoning t h a t precedes Theorem 6.8.1, we see t h a t with every Stone family S of prime GI's of a GBA A one can associate a p-zero-dimensional and p-Hausdorff BS which, by Theorem 6.8.1, is F H P - c o m p a c t if and only if S contains all prime GI's of ,4. The Stone BS of the GBA A is a p-zero-dimensional, p-Hausdorff, and F H P - c o m p a c t BS, which is associated with the Stone family of all prime GI's of S. This BS will be denoted by ( S ( A ) , T1, T2). Theorem
6.8.2. Let A - {A~,A1,VI,~91,1~, ~ ,e,A~,A2, V2,~2} be a GBA, its associated Stone BS, A ( X ) = {A ~ 2 x : A = A1 A A2 or A - A1 U A2, A~ E A ~ ( X ) - 7~ A coTj} and the following conditions hold:
(S(A), T1,7-2) be
(1) the combination of l-closure and 2-interior operators and the l-closure operator as well as the combination of 2-closure and l - i n t e r i o r operators and the 2-closure operator are conjugate over A ( X ) so that ~-2 int T1 C1 A -- T1 C17-2 int T1 cl A and T1 int ~-2 cl A = ~-2 cl T1 int 7-2 cl A f o r each set A c A ( X ) .
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6.8. Bitopological Representation ...
303
Furthermore, prove t h a t A' is G.isomorphic to .4. Since (S(A),7-1,7-2) is the Stone BS of the GBA A, the pair ( f , S ( A ) ) is the perfect G.field representation of A and, by virtue of Theorem 6.7.5, the GFS .F(S(A)), considered as the GBA A(Jz(S(A))) = {fl(a) : a 6 A~},N1, Ul, \,2~, ==~ ,S(A), {f2(b) : b c A~}, N2, U2, \}, is G.isomorphic to A. Thus it remains only to prove t h a t A(.T(S(A))) is G.isomorphic to A'. First, let us show t h a t {fi(a) : a E A~} = ~-i N coTj. By the definition of ~-i, it is obvious that fi(a) c ~-i n co Tj for each element a c A~. On the other hand, let U c ~-i N co~-j be any set. Since the family {fi(a) : a E A~} is the base for 7-i, we have U = U Ut, where tET
Ut c {fi(a) :
a c A~} for each t c T. Therefore by Lemma 3 from [113], U is /-compact since U c coTj and (S(A),T1,T2) is FHP-compact. Hence the /-open covering {Ut : t c T} has a finite subcovering {Ut~: k = 1, n}, t h a t is,
U-
n
U Ut~. But Ut~ c { f i ( a ) " a c A~} for each k c 1, n and since the family
k=l
{fi(a) : a c U c {fi(a) : a the pair h = maps, then it
A~} is closed under the set-theoretic operation union, we obtain c A ~ and hence {fi(a) : a c A~} = ~-i N coTj. Now, if we consider (hi,h2), where hi : {f~(a): a c A~} -+ ~-i N coTj are the identity is obvious t h a t since the GFS's
{{fl(a)'a6
A~Iu{f2(b)'b6A~I,N,U,\,
=~,[]1,[]2}
and
{(TI n COT2)U (T2 n coT1),N,U,\ , ==x(', []i, []~ } coincide, the corresponding GBA's will coincide too. Therefore the GBA's A(~c(S(A))) and A' are the same so t h a t A' is G.isomorphic to A. []
T h e o r e m 6.8.3 (A Generalized Version of Stone's Representation Theorem). Under the hypotheses of Theorem 6.8.2, there exists a one-to-one correspondence (up to a G.isomorphism and a d-homeomorphism) between G B A ' s and p-zero-dimensional, p-Hausdorff and FHR-compact (also called Boolean) BS's such that for every GBA A the GBA
.A(.)~-(S(A)))- {T 1NCOT2, N I , U I , \ , ~ , ==~,S(A),T2 NCOTI,N2, U2,\}, corresponding to the GFS 5(S(A))
\, =<, I11, I
of the associated Boolean BS (S(M), T1, T2), is G.isomorphic to A. Proof. First, assume that A and A' are G.isomorphic. We shall show that the corresponding perfect G.field representations (f, S(A)) and (f', S(A')) are equivalent so that, there exist a bijection h : S(A) + S(A') and a G.isomorphism h* - ( h * I , h ~ ) ' A ( Y ( S ( A ) ) ) ~ A'(gc(S(A'))) such that h~(A) = {h(z) : x c A} for every set By Theorem 6.4.9 the image of a GI under a G.isomorphism preserves prime GI's as well. Ix = ( I ~ , I f ) be the corresponding prime GI.
A c {fi(a) : a c Ai} = ri n c o r j . a G.isomorphism is a GI. Clearly, Let x c S(A) be any point and I f g = (gl,g2) : A + A' is the
G
304
VI. Generalized Boolean Algebra and Related Problems
above-mentioned G.isomorphism, then g ( I x ) = ( g l ( I ~ ) , g 2 ( I ~ ) ) i s the prime GI of A ~. Let the corresponding fixed point from S ( A ~) be the point Yg(lx) and define h: S ( A ) ~ S ( A ' ) in the manner as follows: h(x) = Yg(lx). Clearly, h is a bijection. Let us prove that if B 9{fi(a) : a 9Ai}, then
h~(B) - {h(x) " x
9B }
9{f[(a') " a'
9A'i}.
By virtue of the corresponding definition, we have S ( A ) = {x~: I = (I1,/2) 9S}. Since B C S ( A ) and B 9{f/(a) : a 9Ai}, there exists an element a 9Ai such that
B = fi(a) = { x i : Let B' - { h ( x f ) Therefore
9
xI
a 9
9B}. If gi(a)
hi(B)* - {Y9(I)
(~-CIj, I = (11,/2)
~aVj -
a'
9Ai,' then a'-# 9~(Ii) ,,f----~,a ' Vj' e '
9a'-Egi(Ii) ~
g(I) - ( g l ( i l ) , g 2 ( I 2 ) )
9S}. -# 9y ( I j ) .
a' Vj' O ' ~ g j ( / j ) } ,
E S ' - - { h ( x I ) " xI E B }
and so h~(B) e {f/(a') 9a' c A~}. The pair h* - (h~,h~) is a G.isomorphism since g and f are G.isomorphisms. Indeed, let B' e {f;(a') 9a' e A~} be any set. Then there exists an element a' c A~ such that B' - f[(a'). Hence, there is an element a E Ai such that gi(a) = a ~ since gi are bijections. Let us consider the set f~(a) = B e {f/(a) : a e Ai}. It is obvious t h a t h'~(B) = B' so that h~ are onto. Now we assume that B1, B2 e {fi(a) : a e Ai} and B1 5r B2. Since f~ are bijections, there exist elements hi, a2 c Ai such that a l r a2 and fi(al) = B1, fi(a2) - B2. Let a~ - gi(ai). Then B~ - f~(al) , B; - f~(a~) imply h*(B1) - B~, h~(B2) - B;, and we have h~(B1) r h~(B2) because gi and f~ are bijections. Thus h~ are also bijections. It is trivial to verify the fulfilment of the following conditions: (1) h~(C1 ni (72) -- h~(C1)n~ h~(C2) and h~(C1 Ui C2) - h~(C1)U~ h~(C2) for each pair (C1, C2) 9(Ti n co Tj) X (Ti n co Tj). (2) h~(C1 ni C2) - h~(C1) N{ h~(C2) and h~(C1 Ui C2) - h~(C1)U{ h~(C2) for each pair (C1, C2) 9(rj N cori) x (ri N corj). (3) S ( A ' ) \ h~(C) - h~(S(A) \ C) for each set C 9r~ N co rj. Therefore h* = (h~,h~) is a G.isomorphism and thus ( f , S ( A ) ) and (f', S ( A ' ) ) are equivalent. Together with this equivalence, the definitions of the topologies r~ and r~ immediately imply that the BS's ( S ( A ) , T I , r 2 ) and (S(A'), r~, r~) are d-homeomorphic. This result means t h a t the G.isomorphy of GBA's implies the d-homeomorphy of the corresponding Boolean BS's. Next assume that (X, 71,72) is any Boolean BS satisfying the conditions of Theorem 6.8.2. W i t h (X, 71,72) we associate the GFS
oT'(X)- {(')'1 n co~'2)U (~2 n co')'1), n,u, \, ~, =:~, []1, []2} and its corresponding GBA
r
{~'1 nco~'2, n l , U l , \ , ==xQ,X,~2 n co~'l, n2, u2,\,}.
The GFS 2 ( X ) is the reduced one since (X, 71,72) is p-zero-dimensional and p-Hausdorff. We want to show that the Stone BS (S(A), r~, r2), associated with
6.8. Bitopological Representation ...
305
the Stone family of all prime GI's of the GBA A(a~(X)), is d-homeomorphic to (X, ~1, ")/2).
It is clear that the GFS Jr(X) -- {(~1 (-I CO ")/2) U (~2 A CO ~I) , ['-I, U, \, ==x(, []1, []2}, considered as the GBA
A(.~'(X))
--
{"71
I"'1
CO ")'2, ["ll, U1, \, 23, ==xQ, X , "/'2 f"l co ~1, A2, U2, \ },
is the reduced G.field representation of the same GBA A(.F(X)) in itself. Hence, by virtue of Theorem 6.7.4, with )c(X), considered as a G.field representation of A ( 5 ( X ) ) , one can associate the Stone family of prime GI's of A(Jz(X)). Therefore, applying the reasoning that precedes Theorem 6.8.1 with this G.field representation, we can associate a p-zero-dimensional and p-Hausdorff BS and, as is easy to see, this BS is d-homeomorphic to (X,'71,'72). But the BS (X,'71, ~72) is FHP-compact and, by Theorem 6.8.1, the corresponding Stone family S contains all prime GI's of A. Hence we conclude that (X, ~/1, ~/2) is d-homeomorphic to the Stone BS (S(A), T1, T2). The rest of the proof follows directly from Theorem 6.8.2. r-] D e f i n i t i o n 6.8.4. A G.lattice Z; = {L1,A1, Vl,@, m--X,e, L2, A2, V2} is said to be G.complete if its every subset has an ic-infimum and an i,~-supremum. Now we say that a GBA A = { A 1 , A 1 , V I , q P l , O , 4 , e , A2, A2, V2, qP2} is G.complete if the corresponding G.lattice {A1, A1, V1, 0, 4 , e, A2, A2, V2} is G.complete. L e m m a 6.8.5. Let f = (fl, f2)
: A
and A', where A is G.complete. Then
--+
G
A t be a G.isomorphism of GBA's A
fi ( i , - i n f B) - i , - i n f (fl ( U l ) U f2 (U2) ),
for every subset B c A1 U A2, where B rq A1 = B1 7k 23 7L B2 = B r~ A2, and, therefore, A' is also G.complete. Pro@ It suffices only to prove the equality fi(ic-inf B) = ic-inf(/1 (B1) U f2(B2)) since the proofs of both cases are similar. Since A = {A1,/~l,Vl,~91,l~,4 ,e, A2, A2, V2, qp2} is G.complete and f = (f~, f2) is a G.isomorphism, it is clear that A~ = {A~, A,, V,, ~&, (3, <_, e} are complete BA's and the fact that
f~. A~ - {A~, A,, v,, e,, e, _<, ~} -~ A'~ - { < , A,,l v,,l e,,l e , l ~I , (31} i
are isomorphisms implies that A{ -
i i I I I ~ i {A{, A~, V~, ~,, (3, , e'} are also complete i
BA's. Therefore L(infB~) = infL(B~), and it remains to show that (1)-(3) of Definition 6.1.6 hold for fi(i~-inf Bi) when they hold for ia-inf Bi.
306
VI. Generalized Boolean Algebra and Related Problems
(1) Since A is G.complete, for a subset B C A1U A2, there exists an ic;-inf B. Let b - f~(i~;-infB). Then by (1) of Definition 6.1.6, we have i~-infB <_ inf B~ and ic-inf B 4 inf Bj. Therefore b <_' fi(inf Bi) - inf fi(Bi) and b4'fj(inf By) - inf fj(Bj) i so that (1) holds. (2) Let x E A{, z _<' inffi(Bi) and x4'a for each a E fj(Bj). Since f i (fl, f2) is a G.isomorphism, that is, fl and f2 are isomorphisms, we conclude that f - 1 _ ( f l 1, f~-l) is also a G.isomorphism and, therefore,
f/-l(x) ~ f[-l(inf fi(Bi)) -- infBi, f/-l(x) ~ fj-l(a) i
for each element fj-l(a) c Bj. By (2) of Definition 6.1.6, f / - l ( x ) ~_ ic-infB and i thus x _<' fi(i~-inf B). i (3) Let x c Aj,r Xr 4 a for each a c fi (Bi) and x _<' inf fj (Bj). Then we easily J obtain fj-l(x) _< f 7 1 ( i n f f j ( B j ) ) - i n f B j , fj-l(x) 4 f ( l ( a ) J for each f ( l ( a ) c Bi. Hence, by (3) of Definition 6.1.6, fj-l(x) 4 iG-inf B and thus x4' fi(ia-inf B). Now, Definition 6.1.6 implies that fi(ic-inf B) -iG-inf(fl(B1 ) U f2(B2)). It is clear that if B c_ Ai, then, we have fi(i~;-infB) - i~-inff(B) and fi(ic-su p B) - / c - s u p f(B). [:] Our next theorem converts the property of a GBA .4 being G.complete into the property of the Stone BS (S(N), rl, r2) being p-exremally disconnected. Theorem 6.8.6. Under the hypotheses of Theorem 6.8.2, { A 1 , A1, V1, ~1, 1~, 4 , e, A2, A2, V2, ~2} is G.complete if and only
a GBA
A
-
if (S(A), rl, r2)
is p-extremally disconnected. Proof. First, we assume that A is G.complete. Then, by Theorems 6.8.2 and 6.8.3, the GBA r
{T1 n COT2, N I , U I , \ , { ~ , ==~,S(ce),T2 n COTI,N2, U 2 , \ } ,
which corresponds to the GFS .~'(S(A)) -
{(TI NCOT2) U (T2 NCOTI),N,U,\ , = = x ( , [ ] l , [ ] 2 } ,
is G.isomorphic to A. Hence, by virtue of Lemma 6.8.5, for every family of /-open and j-closed subsets of (S(A),rl, r2), there exists a smallest /-open and j-closed set containing all of them. Let U c r~ \ {~} be an arbitrary set. Since p-ind S(A) - 0, for each point x E U, there exists an/-open and j-closed neighborhood U(x) such that U(x) c U so that U - U U(x). Therefore for the
xEU
families of/-open and j-closed sets {U(x) 9 x EU}, there exists a smallest/-open and j-closed set U* such that U c_ U*. Thus rj cl U - r~ int rj el U - U* and, consequently, (S(A), rl, r2) is p-extremally disconnected.
6.8. Bitopological R e p r e s e n t a t i o n . . .
307
Conversely, let (S(A),T1,T2) be a p - e x t r e m a l l y disconnected BS and B c A 1 0 A2 be any subset. Let us prove t h a t there exist i ( j - i n f B and i c - s u p B . W i t h o u t loss of generality, we consider the case of i~-sup B. We have
B A A1 = B1 ~ ;g ~ B fB A2 = B2. Clearly, f l ( B 1 ) U f2(B2) C S ( A ) since f = ( f l , f 2 )
: A --~ A ( ) c ( S ( A ) ) ) is G
G . i s o m o r p h i s m and u
-
9
a
u
a
9
where {fi(a)
9 a ~Bi} c ~-i ~ c o T j
and { f j ( b ) "
b ~ By} c rj ~ c o T i .
It is obvious t h a t
U fi(a)-UEz-i, U fj(b)-VETj
aEBi
bEBj
and
Tj cl U = T~ int Tj cl U c T~ N COTj , T i c l V = v j i n t T ~ c l V c T j A c o T i since ( S ( A ) , T1, T2) is p - e x t r e m a l l y disconnected. Therefore, there exist elements a c Ai and b c Aj such t h a t f~(a) = Tj cl U, fj(b) = ~-~cl V and 7-i cl V Ui Tj cl U = [Ti cl V]~ Ui Tj cl U = 7-i int Tj cl Ti cl V Ui Tj cl U = 7i int Tj cl (~-~int Tj cl 7-i cl V U Tj int 7-i cl Tj cl U) = ~-i int (Tj cl 7-i int Tj cl 7-i cl V U Tj cl Tj int ~-i cl Tj cl U) c ~-i A co Tj since Tj cl ~-i int Tj cl Ti cl V U Tj cl Tj int 7-i cl Tj cl U C co Tj and ( S ( A ) , T1, T2) is p-extremally disconnected. Therefore T~ cl V U~ rj cl U c ~'i ~ co Tj is the smallest set containing fi (Bi) U f j (Bj) and so Ti cl V U Tj cl U = / ( ; - s u p ( f i ( B i ) U f j ( B j ) ) . By L e m m a 6.8.5, there exists i(.-sup B - f / - l ( i a - s u p ( f i ( B i ) U f j ( B j ) ) ) since f - 1 = (f~-l, f~-l) is also a G . i s o m o r p h i s m and hence ,4 is G.complete.
D
T h e o r e m 6 . 8 . 7 . For any G.component E of a G.complete G B A A = {A1, A 1 , ,e, A2, A2, V2, p2}, we have ai(L) = E = hi(R), where a = 1 , - s u p E and b = 2,:-sup E, that is, a = b V1 (~ and b = a V2 0 .
VI,~I,(:~),~
Pro@ Let E = E1 U E2 C A1 U A2 be a G . c o m p o n e n t , t h a t is, E = C d , (Cd(;E). Clearly, there exist a = 1 , - s u p E and b = 2(~-sup E. Then, on the one hand, by (5) of T h e o r e m 6.2.21, we have A z = C d , ( C d ( ; E ) and on the other hand, by (1) and (2) of T h e o r e m 6.2.22, we have aI(L) = A z = Cd~ (Cd(:E) = E = hi(R). T h e rest is obvious.
If]
308
VI. Generalized Boolean Algebra and Related Problems
R e m a r k 6.8.8. Let .4 = {A1, A1, Vl, (~91, t~, ~--~,e, A2, A2, V2, ~2} be a G.complete GBA, B = B~ U B2 C A1 U A2 and ~ ( g ) -- ~ I ( B 1 ) U ~ 2 ( B 2 ) ,
~)(B) = ~ ) l ( g l ) U @2(B2),
where i~,-infB-(
Aj x) Ai( Ai y) and / a - s u p B - ( x6 B 3
y 6 B~
Vj x) Vi( Ai y). xE B j
y ~ B.~
Then it is not difficult to see that
~( a~ x ) - v3 ~(x), ~( v~ x ) - a~ ~(x), x 6 B,,
x ~ B.i
~( /~ x ) x 6 B~
x ~ B,
v~ V~(x), r
x 6 B~
x 6 B~
v~ x ) -
x 6 B,i
/~ V~(x)
x 6 B.,
and, therefore, ic-infB-(
Aj x) Ai ( Ai y ) - x6B 3
= ~j((
y6Bi
Vi pj(x)) Vj ( Vj Pi(Y))) - ~ j ( j a - s u p ~ ( B ) )
x 6 B:i
y6 B i
so t h a t / a - s u p B = pj(jG-inf ~(B)). Similarly,
i~;-inf~ - ( =r
A~ x)A~ ( A~ ~ ) -
xEBj
Vj ~ j ( x ) ) V i ( Vi r xCBj
yCBi
~i(ie~-sup~(B))
y6Bi
so t h a t / a - s u p B = ~i(iG-inf ~(B)). The proof of Corollary 6.8.12 below will be facilitated by Remark 6.8.8. D e f i n i t i o n 6.8.9. A GBA A = {A1, A1, V1, Pl, (9, 4 , e, A2, A2, V2, ~2} is said to satisfy the countable G.chain condition if every G.disjoint system of non-zero elements of A1 U A2 is countable. T h e o r e m 6.8.10. A GBA .4 = {A1,A1,V1,q~)1,0,4,e, A2, A2, V2,~2} satisfies the countable G. chain condition if and only if every set E = E1 U E2 c A1 U A2 has a countable subset D such that u a (D) = % (E).
Proof. First, let B = B1 U B2 C A1 U A2 be a G.disjoint set (system) of non-zero elements of A1 U A2, D c B, IDI <_ b~0 and u c (D) = u G(B). If, for example, there is an element x c (B \ D) N A~, then d A~ x = (9 ~ d 4 ~ ( x ) for each d E D and hence pi(x) c % ( D ) = u~ (B), which is impossible since x 4 pi(x) is false. Thus B = D. To prove the converse, let the countable G.chain condition be satisfied and E = E1U E2 c A1U A2 be an arbitrary subset. If I = (I1, I2) is the GI generated by E, then u o ( E ) = u , ( I 1 U h ) . Indeed, always u~,(I1 U h ) c % ( E ) and if, for example, x c u ~ ( E ) N Ai, then y 4 x for each y c E. Let z c Ii be an arbitrary element. Then, according to Theorem 6.2.5, there are sequences al, a 2 , . . . , an C Ei and bl, b2,..., b,~ c Ej
6.8. B i t o p o l o g i c a l R e p r e s e n t a t i o n . . .
309
such t h a t z < al Vi a2 V~... V~ a~ and z ~ bl Vj b2
Vj ...
b,~.
Vj
Clearly, ak <_ x for each k - 1, n and bz ~ x for each 1 - 1, m so t h a t alvla2v1.-.vla~<_x,
bl Vj b2 Vj . . . Vj b,~ 4 x
i
and, therefore, z <_ x. But z c Ii is an a r b i t r a r y element and hence x c uc;(IltJI2), i
t h a t is, uc; (E) - uc; ((11 CJ/2). Furthermore, suppose t h a t 13 - { B ~ - B ~ U B ~ } ~ c T is a family of all G.disjoint sets consisting of non-zero elements such t h a t Ba C 11 [J/2 for each a E T. It is evident t h a t the family B is partially ordered by inclusion, t h a t is, Ba < BZ ~ B~ c B~ and if/C c 13 is linearly ordered, then ]C is b o u n d e d from above by the set B [.J Ba. It is not difficult to see t h a t B is a G.disjoint set B~6/E
of non-zero elements. Therefore, by Zorn's lemma, 13 contains a m a x i m a l element, say B0 - B ~ U B ~ t h a t is, a maximal G.disjoint set of non-zero elements. Since B1~ ~ C I1 U / 2 , we have u c ; ( I i U I 2 ) c uc;(Bo). On the other hand, let, for example, z C u ~ ; ( B o ) A A j , t h a t is, x ~ z ~ xAi~j(z) -- 0 for each x c B ~ ~ If z-Eu(;(I1 U I2), then, for example, there is an element y c I~ such t h a t y ~ z is false ~ ao - y A ~ j ( z ) ~ @. T h e n x A i a o -- @ since ao < ~ j ( z ) , where i
x C B ~ U B ~ is an a r b i t r a r y element.
Moreover, ao < y implies ao c I~ since i
I - (I1,I2) is a GI. Hence, we have found the element ao c Ii \ {@} such t h a t x Ai ao -- @ for each x c B1~ U B ~ This contradicts the m a x i m a l i t y of Bo and thus uc;(I1 U 12) - uc;(Bo), where Bo < Ro since the countable G.chain condition is satisfied. Clearly, E U Bo c I1 U/2. But we would like to find a set D c E such t h a t ]D[ < Ro and u~ (D) - u , (E). If x c Bo N A~ is an a r b i t r a r y element, t h a t is, if x ~ I~, then there are elements al,a2,...,a~
~ B~
bl,b2,...,bm
~ B~
such t h a t x < al Vi a2 V i - . . V~ a~ and x ~ bl Vj b2
Vj
bin.
".. Vj
i
Suppose t h a t D x --
{{al,a2,...
, an; bl, b2, . . .
,bin}
o
x <_ al Vi a2 Vi i
o
o
i
Vi an
and x ~ bl Vj b2 Vj . . . Vj bm }. Then [Bo[ _< Ro implies [D] _< Ro, where D -
U
Dx and u c ( D ) - uc~(B0 ) - uG(/1
U
12) - u(;(E).
[-1
x E B oi U B 2o
D e f i n i t i o n 6 . 8 . 1 1 . A G B A A = {A1,A1,VI,(~I,(~,4 ,e, A2, A2, V2,(P2} is said to be a G.Boolean a-algebra if its every countable subset has an ic-infimum (and, therefore, of course, and i , - s u p r e m u m ) .
310
VI. Generalized Boolean Algebra and Related Problems
It is clear that the notion of a G.Boolean a-algebra is an intermediate concept between that of a GBA and a G.complete GBA. C o r o l l a r y 6.8.12. A G.Boolean a-algebra .4 = {A1, A1, V1, ~ 1 , (~, 4 , e, A2, A2, V2, ~2} which satisfies the countable G.chain condition, is G.complete.
Pro@ Indeed, let B = B1 U B2 c A1 U A2 be any subset. Then according to Theorem 6.8.10, there exists a subset E = E1 U E2 c B such t h a t [E I < R0 and u(:(E) = u , ( B ) . Thus i(~-supE = / , - s u p B and by Remark 6.8.8, i(~-infE =
i~-inf B.
[-1 6.9. G e n e r a l i z e d B o o l e a n R i n g s
The notion of a G.Boolean ring is a special G.ring version of the notion of a GBA. D e f i n i t i o n 6.9.1. A G.ring is a non-empty set R - R1UR2 together with four binary operations | @i : (Rk x Ri) -~ Ri called G.multiplication and G.addition, respectively, which satisfy the conditions below: (1) There exists a unique element O E R1 N R2, called the zero element, such that x @~ (9 = (9 @i x = x for each x c Ri. (2) To each x E R~ there corresponds a unique element c~(x) c Rj such that x |
c~(x) = e .
(3) G.addition is G.commutative: xO~y=yOix
if (x,y) E R i x R i
(x@iy)-(y@jx)@iO
and
if (x,y) E Rj x Ri.
(4) G.addition is G.associative:
(x |
y) |
z = x e~ (y |
~)
if x, y, z E Ri or x E Rj, y, z E Ri and
(x e~ y)|
~ = x e~ (y e~ ~)
if x, y c Rj~ Z C R i o r x, z C R i , y E Rj. (5) G.multiplication is G.associative:
(x |
y) |
~- x|
(y |
z)
if x, y, z E Ri or x E Rj, y, z E Ri and
(~ |
y)|
z = x |
(y |
z)
if x, y E Rj, z E Ri or x, z E Ri, y E Rj. (6) G.multiplication is G.distributive with respect to G.addition:
x |
(y |
z) = (~ |
y) |
(x |
z)
y) |
(~ |
~)
(y |
z)
if x, y, z E R~ or x E Rj, y, z E Ri,
x | if
x, y
c Rj,
(y e~ z) = (x |
Z C R i o r x, z C R i , y
(x |
y) |
~ = (x |
c Rj, ~) |
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6.9. G e n e r a l i z e d B o o l e a n Rings
317
(2) hi o Oi - ED~ o (hk, hi) so that hi(x Qi Y) - hk(x) O'i hi(y) for each pair (x, y) e Rk x R~. (3) c', o hi - hj o ci so that c{(hi(x)) - hj(ci(x)) for each x 9Ri. A G.homomorphism h = (hi,h2) of GBR's is a G.isomorphism if hi and h2 are bijections. It is not difficult to ascertain that the following statement is true. T h e o r e m 6.9.7. Let A be a GBA, 7g(A) be the GBR, constructed from A by the rules stated in Theorem 6.9.4, and let A(Tg(A)) be the GBA constructed from 7g(A) by the rules stated in the same theorem. Then A is G.isomorphic to A(Tg(A)) in the sense of Definition 6.4.6. Similarly, ifTg is a GBR, A(Tg) is the GBA, constructed f f o m 7g, and 7~(A(7~)) is the GBR, constructed from A(Tg), then 7g is G.isomorphic to 7g(A(Tg)) in the sense of Definition 6.9.6.
Generalized B o o l e a n Algebras and R e l a t e d Problems. R e p r e s e n t a t i o n T h e o r e m s From the late 19th century the tendencies enabling a scholar to view different mathematical theories from the unified standpoint began to gain prevalence and reached rather a high level of development. This observation is confirmed by many studies, of which, keeping in mind our interest, we would like to mention M. H. Stone's remarkable theory of representations that brought together the plain algebraic structures of Boolean algebras with topological type structures, that is, with a special class of TS's [246], [247]. For quite a long time the proofs of theorems had been based on the Boolean algebra laws. G. Boole was the first to formalize these laws [36], [37], thereby providing a powerful incentive to the development of symbolic logic [28]. Subsequently, Boolean algebras became widely adopted in such areas as probability theory, functional analysis, topology, Stone's representation theory, and others. Chapter VI continues and develops the study begun in [91], [95], [100]. If, presumably, the definition of a Boolean algebra employs the notion of a partially ordered set (briefly, poset), whereas a general definition of a TS is based on the notion of a metric, then by considering a more general situation with quasi metrics one may obtain the notion of a BS. A further development of the theory of BS's made it possible to introduce and study an algebra of a new type that is based on a nonordinary variant of a quasi ordered set and the corresponding representation of which brings one to BS's. We believe that this algebra, which we call a generalized Boolean algebra, is interesting not only as the subject of independent research, but also can be used as an important tool in establishing its various relationships with other areas of mathematics. Moreover, since a quasi ordered set of special type, which is the union of two posets forms the basis for defining both generalized Boolean algebras and double Boolean algebras [134], [261] and the latter algebras have already been used in metamathematical studies, due to this common basis, generalized Boolean algebras can also be used in mathematical logic. In Sections 6.1-6.6 we introduce and study generalized Boolean algebras and some other related important questions. To this end, we first define a generalized ordered set (briefly, goset) in terms of a quasi ordered set which is a union of two posets whose partial orders are induced by the quasi order relation and whose intersection contains only zero and unit elements. Furthermore, to define a generalized lattice, for any subset of a goset we introduce the important notions of i~.-inf and i(;-sup. A different approach to defining a generalized lattice as 193
194
VI. Generalized Boolean Algebra and Related Problems
an algebra, satisfying in particular, the generalized associativity and generalized absorption laws, is also used. Theorem 6.1.15 shows how we can pass from the generalized lattice, defined by means of a goset, to the generalized lattice using an algebra and vice versa. The results obtained make it possible to connect the posers, making up the goset, by the one-to-one correspondence and thus to introduce the basic notion of a generalized Boolean algebra. The one-to-one correspondence between the posers is determined by the generalized identity operator which plays an important role in our further studies because, on the one hand, it determines the interdependence of two structures forming a generalized Boolean algebra and, on the other hand, allows us to characterize a generalized Boolean algebra in a different way. We also discuss some other operators and binary operations and give a few interesting examples of generalized Boolean algebras. Furthermore, we assign the V-formation and the corresponding strong amalgamation to every generalized Boolean algebra, and we define a generalized quasi measure on a generalized Boolean algebra and a bitopological generalized Boolean algebra. In Sections 6.2 and 6.3, we obtain key generalizations of an ideal and its variety such as a prime, that is, a maximal generalized ideal, left and right principal generalized ideals, and generalized ideals generated by different pairs of families of sets. We introduce and investigate the notion of a generalized Stone family of prime generalized ideals as well as the notion of a generalized dual to a generalized ideal, that is, the notion of a generalized filter together with its variety. The relation between generalized components and generalized ideals is also established. Other results obtained in the first half of this chapter concern the important notions of generalized homomorphic and generalized isomorphic maps, (i,j)-atoms, and p-atomic generalized Boolean algebras and generalized Boolean factor algebras. The remaining three sections deal with the bitopological modification of Stone's representation theory, generalized complete generalized Boolean algebras, and generalized Boolean rings. We introduce the notions of a generalized field of sets, its reduced version, and generalized field representation. Based on the results of Section 6.2, it is proved, in particular, that there exists a one-to-one correspondence between the reduced generalized field representations of a generalized Boolean algebra .4 and the generalized Stone families of prime generalized ideals of ,4. Thus every generalized Boolean algebra becomes generalized isomorphic to the reduced generalized field of sets. With every generalized field representation of a generalized Boolean algebra we associate a BS and give the necessary and sufficient conditions under which this BS becomes FHP-compact. Our main result is the generalized version of Stone's representation theorem which states that under special hypotheses there exists a one-to-one correspondence between generalized Boolean algebras and p-zero dimensional, p-Hausdorff and FHP-compact (also called Boolean) BS's. We would like to recall that an attempt at connecting the bitopological structure with a generalized algebraic structure was for the first time made in [91]. Throughout this chapter, the term "generalized" will be denoted for brevity by the symbol "G" and k, 1 c { 1, 2} , k ~r 1.
6.1.
Gosets,
Generalized
Lattices,
...
195
6.1. Gosets, Generalized Lattices, Generalized Boolean Algebras, and the Corresponding Operations Definition 6.1.1. A quasi ordered set (P, 4 ) is said to be a goset if P P1 U P2, where (P~, <) are nonempty posets and the restrictions 4 ]P~ -<_. i
i
It is clear that every poset is a goset, but not conversely.
Example 6.1.2.
(X, T1,T2) be
Let
a nonempty BS.
Then the families
j-OZ)(X) r ;g # (i, j)-OT)(X) are partially ordered by the set-theoretic operation inclusion and the binary relations, defined on the sets
P - j-oz)(x)
o (i, j ) - o z ) ( x )
as follows A1, A2 c P, A1 4 A2 <--> Aa c_ ~-j clA2 are the relations of quasi order on P. Moreover,
(P - j - o z ) ( x )
o
j)-oz)(x), 4 )
are gosets. Indeed, by (2) of Proposition 1.3.10, the restrictions 4 A1 4 A2 ~
I(i,j)-oz)(x) give
A1 C_ Tj clA2 <---> Tj clA1 C_ Tj clA2 <----> A1 C_ A2
(i,j)-oz)(x) =c_=<. Similarly, by the topological correspondence of (2)
and so ~
of Proposition 1.3.10, 4
i
Ij-oz)(x) =c_=<_. J
The quasi order 4 , defined in Example 6.1.2, shows that (P (1, 2)-OD(X), 4 ) in Example 1.3.5 is a goset. For the sets
-
2-or(x)u
{a, b} c 2-OT)(X) and {at c (1, 2)-OT)(X) we have {a,b} 4 a and {a} 4 {a,b}, but {a,b}-r {a}. Thus t h e g o s e t (P, 4 ) is not, generally speaking, a poset. If (P, 4 ) is a goset, then for any pair x, y c P the notation x -~ y means z 4 Y and z -r y. Therefore for z c P1 \ P2, Y c P2 \ P1, we have z -< y <---> x 4 Y, whereas for z, y c Pi, we have z -~ y <---> z < y. i
Therefore in Example 1.3.5, we have {a, b} -~ {a} and {a} -~ {a, b}. e b
f
.. -TPx, --- -'- -'-
a
// J
d
f
1 1
Diagram 1
In
Diagram
goset (P = {a,b,c,d} and Pe - {c,e,f}, the small circles denote the elements; the circles, corresponding to the elements x, y c Pi, are connected by the solid line while for x c Pi and y c PO, the connection is broken.
{a,b,c,d,e,f},4
1
of
the
), where P1
196
VI. Generalized Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
The quasi order r e l a t i o n x 4 y for x E Pi, y E Pj means that y is over x or x and y lies on the same horizontal line; whereas the restriction 4 IP, - < gives that x < y if y is over x or x - y. i
It clearly follows that for a E P1, f E P2, we have a -< f and f -< a, a ~ f. Note that in the sequel, the indices i~; and Ja satisfy as usual the conditions i a , j a C {1,2}, i~ =fi JG, and, moreover, accentuate a more general character of the corresponding notions as compared with usual structural ones. D e f i n i t i o n 6.1.3. Let (P, 4 ) be a goset and A c_ P be a subset such that A N P1 - A1 ~ ~ ~ A2 - A N P2. Then A is called to be a G.chain (G.convex set) if a 4 b o r b 4 a for each pair a, b E A ( a , b c A a n d a 4 c4 bimplycEA). If A c_ P is a G.chain (G.convex set), then A~ are chains (convex sets) in the Usual sense.
An element xE Pi is an ia-lower (/,-upper) bound of a subset A - A 1 U A 2 c_ P, where ( P - P 1 U P 2 , 4 ) isagoset, ifx <_aforeachac Ai a n d x 4 b f o r e a c h i
b E A j (a <_ x for each a c Ai and b 4 x f o r e a c h b c A j ) . i The sets of all ia-lower (/,-upper) bounds of A are denoted by i a - l ( A ) - { x C Pi " x < a, a c Ai and x 4 b, b c A j } i
(ia-u(A)
- { x E Pi " a <_ x, a c Ai and b 4 x, b E A j } ) . i
Moreover, l~;(A) - I ~ - / ( A ) U 2 , - l ( A ) - { x E P " a ~ x, a E A } (ua(A)
- l a - u ( A ) U 2 a - u ( A ) - {x C P " x 4 a, a C A } ) .
Furthermore, if (P, ~ ) is a goset, then a subset A c_ P is G.normal (or A is G.normally included in P) if a c A, x r P and x 4 a imply x r A. Hence, if A is G.normal, then l~; (A) c A. D e f i n i t i o n 6.1.4. An element x c Pi is called an /a-smallest (/a-largest) element of the goset (P - P1 U P2, 4 ) if x is the smallest (largest) element of the poset (Pi, <) such that x 4 a (a 4 x) for each a c Pj. i
The i~;-smallest element (if it exists) will be denoted by the symbol ic-@ (i~-zero element), while the io-largest element (if it exists) will be denoted by the symbol i~jm (iG-unit element). Since the existence of smallest (largest) elements of posets (P~, <) is the neci
essary condition for the existence of i a-smallest (i~-largest) elements of a goset ( P - P1 UP2, 4 ), the uniqueness of/a-smallest (i(~-largest) elements (if they exist) is clear. In the context of the arguments presented above let us consider the following elementary, but necessary, example.
6.1. G o s e t s , G e n e r a l i z e d
Lattices,
...
197
E x a m p l e 6.1.5. Let the set P - { - 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4 } be linearly ordered (by the usual order) and, therefore, quasi ordered as the set-theoretic union of linearly ordered and thus partially ordered sets P1 - { - 4 , - 3 , - 2 , - 1 } , P2 - {1, 2, 3, 4}. Clearly, ( P - P1 U P2, 4 ), where x 4 y ~ x <_ y for each pair x, 9 c P and for the usual order <, is a goset. The element 1 c P2 is the zero element of P2, but it is not the 2a-zero element of P, while - 4 c P1 is the l a - z e r o element of P. Similarly, 4 c P2 is the 2a-unit element of P, while - 1 c P1 is the unit element of P1, but it is not the 1a-unit element of P. In the sequel we shall consider, in general, gosets ( P - P1 U P2, 4 ) such that (P1, <_) and (P2, <) are posets with common zero and unit elements denoted by G 1
2
and e, respectively; obviously, then O - l a - O - 2a-(~ and e - l a - e - 2a-e. Now we introduce the notion which is highly important for our later considerations. D e f i n i t i o n 6.1.6. Let (P, 4 ) be a goset and A c_ P be a subset such that A1 r 2~ r A2 - A N P2. Then an i(;-infimum of A (briefly, i~;-inf A) (an i , - s u p r e m u m of A (briefly, i , - s u p A ) ) is an element from i(;-l(d) ( i , - u ( A ) ) , satisfying the conditions below: A n P1 -
(1) i(;-inf A < _ inf A~ (sup A~ < / , - s u p A) and i
i
i(~-inf A 4 inf Aj
( sup Aj 4 i , - s u p A ) .
(2) I f x c P~, x <_infA~ (supA~ <_x) a n d x 4 a (a 4 x) for e a c h a c Aj, i
then x _< i , - i n f A (iG-sup A < x). i
i
(3) I f x c Pj, x 4 a (a 4 x) for e a c h a c A i then x 4 i(;-inf A ( / , - s u p A 4 x).
andx
(supAj < x), 5
It is obvious that the existence of inf A1 and inf A2 (sup A1 and sup A2) is the necessary condition for the existence of i , - i n f A ( / , - s u p A). Moreover, the notions of i~;-inf A (i~-sup A) will be meaningful if and only if their uniqueness is proved when they exist. As an example, let us consider the case of i~-sup A. If we assume that x and y are both i~;-sup A, a - sup Ai c Pi and b - sup Aj c Pj, then by (1) of Definition 6.1.6, we have a <_ x, b 4 x and a < y, b 4 y. It is obvious that z 4 x for i
each z E Aj since z <_ b ~
i
z 4 b. Therefore x E Pi, a < x and z 4 x for each
2
i
z c Aj. But y - / ( ; - s u p A and from (2) of Definition 6.1.6 it follows t h a t y <_ x.
i On the strength of the same argument, we conclude t h a t x _< y and thus x - y. i
R e m a r k 6.1.7. Let (P, 4 ) be a goset and A c_ P be a subset such that A n P1 - A1 7s 2~ r A2 - A n P2. First, we shall show t h a t j~;-inf A 4 i(;-inf A and j(~-supA 4 /(;-supA. Our consideration involves only the case of infimum since the case of supremum can be proved similarly. By (1) of Definition 6.1.6, jc;-inf A 4 inf Ai. But inf Ai < x ,e---->, inf A~ 4 x for each x E A~. Therefore i
198
VI. Generalized Boolean A l g e b r a and R e l a t e d P r o b l e m s
jc-inf A 4 x for each x E Ai. Further, by (1) of Definition 6.1.6, jc-inf A < inf Aj.
7
Since jG-inf A c Pj, jc;-inf A 4 x for each x c A~ and jG-inf A < inf Aj, by (3) of
7
Definition 6.1.6, we have j~-inf A 4 ia-inf A.
Moreover, for A=A~ c_ Pi, Definition 6.1.6 reads as follows: i~-inf A (i~-sup A) is an element inf A E Pi (sup A c P~) such that x E Pj and x 4 a (a 4 x) for each a E A imply x 4 inf A (sup A 4 x). Without loss of generality, let us consider the case o f / , - s u p A. Since A - Ai, we have Aj c_ Ai. By (1) of Definition 6.1.6, sup A _< i~-sup A. On the other hand, assuming that x c Pi and sup A _< x, we obtain
i
i
supAj <_ s u p A since Aj c_ Ai - A c_ Pi. i Besides, sup A < x implies i sup Aj <_ x and so a <_ x for each a E Aj, i
that is a 4 x for each a E Aj. Therefore xCPi,
s u p A _ < x implies a 4 i
x for each a c A j
and, by (2) of Definition 6.1.6, we have ia-sup A _< x. Hence sup A <_ x implies i~;-sup A _< x for each x E Pi i
i
and, therefore, i(~-sup A < sup A i
so that i ( ; - s u p A - s u p A.
Finally, assuming that x c Pj and a 4 x for each a c A, we obtain a 4
x ~
a < x
for each
a
E
Aj
J
since Aj c_ A and hence sup Aj _< x. Thus J xcPj, a 4 x for each a c A
imply s u p A j _ < x J
(since Aj c_ A - Ai) and, by (3) of Definition 6.1.6, i~;-supA 4 z. But, as we have seen above, i a-sup A - sup A and, therefore, x c Pj, a 4 x for each a c A imply sup A 4 x. Clearly for A c_ Pi, the existence of inf A (sup A) is the necessary condition for iG-inf A (it-sup A) to exist and, therefore, iG-inf A (it-sup) (if the latters exist) are unique by virtue of the uniqueness of inf A (sup A). Nevertheless we want to show by giving a simple example below that the existence of inf A (sup A) is not sufficient for i~j-inf A (iG-sup A) to exist. E x a m p l e 6.1.8. Let X - {a, b, c, d, e}, ~-1- {2~, {a}, {b}, {c}, {d}, {a, b}, {a, c},
{a, d}, {b, c},{b, d},{c, d}, {a, b, c},{a, b, d},{a, c, d},{b, c, dI,{a, b, c, d},{a, b, c, e}, X}, and 7-2 be the discrete topology on X. By Example 6.1.2, ( 1 - O r ( X ) U (2, 1)-OZ)(X), 4 ) is a goset.
6.1. Gosets, Generalized Lattices, ...
199
Let us consider the family of sets ~41 = {{a, d}, {b, d}} C 1-Or(X) and the set {d, e} c (2, 1)-OZ)(X). Then {d, e} = {a, d, e} n {b, d, e} = 7-1 cl{a, d} n 7-1 cl{b, d} so that {d,e} ~ {a,d}, {d,e} ~ {b,d}. But inf .At -- {a, d} n {b, d} = {d} E co 7-1. Therefore {d, e} 4 inf A1 does not hold, that is, 1a-inf.41 does not exist. Of considerable importance is E x a m p l e 6.1.9. Let (X, 7-1,7-2) be a nonempty BS. As is well known, 7-i with the set-theoretic operation inclusion as partial orders are complete distributive lattices with respect to the lattice operations
AUs-7-iintNUs and V Us- Uus sES
sES
sES
sES
for every subfamily N - {Us}s~S c_ 7-i. One can easily satisfy oneself that the maps T~ int 7-j c l ' T i ~ ~-i have the following properties: (1) U C_ 7-i int 7-j cl U for each U c Ti. (2) If U c_ V, then T~ int 7-j cl U c_ 7-i int Tj cl V for each pair U, V c 7-i. (3) 7-~int 7-9 el 7-i int 7-r cl U - 7-i int 7-j el U for each U c 7-i. Thus 7-i int 7-j cl are closure operators on 7-i in the sense of [105, p. 13]. Moreover, the "closed" (in the sense of such operators) elements of 7-i are the (i, j)-open domains of (X, 7-1,7-2), and by Theorem 8.1 from [105], the families (i,j)-(.gz)(X) form complete lattices with the same partial order as that of 7-i, while by Theorems 8.1 and 8.2 from [105] the lattice operations in (i,j)-O~?(X) are written aS
A' Us --7-iint
~') Us
sCS
sES
and V' Us --7-,int7-jcl U Us sES
sES
for every subfamily b / - {Us}scs C (i,j)-O2P(X). Note that the latter equality V-s sES
d. U sES
is actually the essence of Lemma 6.1.2, proved in [2381. As Example 6.1.8 shows, for a BS (X, 7-1 < 7-2) and the goset (1-OZ)(X)U (2, 1)-OZ)(X), 4 ), there does not always exist l c - i n f A, where A c 1-O~P(X) is some subfamily. Furthermore, in that case for the BS (X, 7-1 < 7-2) neither does there always exists l c - i n f A for a subfamily .4 c 1-O79(X)U (2, 1)-OZ)(X) such that A N 1-OZ)(X) - J[1 ~ ~ r ~ 2 - - W ~ A (2~ 1)-OZ)(X). Indeed, if we assume that l a - i n f A exists for any such A and take the BS from Example 6.1.8 as a BS (X, 7-1 < 7"2) and the family {{a, d}, {b, d}} U (2, 1)-O:D(X) as A, then by virtue of the second part of Remark 6.1.7, there must exist l c - i n f A1, where A1 - {{a, d}, {b, d}} c 1-OD(X), which contradicts Example 6.1.8. Therefore ic-inf A, where .4
n 1-OT)(X)
- A1 -~ 2~ r A2 - A
n (2, 1)-OT)(X),
200
VI. Generalized Boolean Algebra and Related Problems
does not always exist for a BS (X, 7"1 < 7"2). However, we shall show below that for a BS (X, 7"1 < 7"2) and the goset (1-O2)(X) U (2, 1)-OD(X), 4 ) there always exist 1a-sup A, 2c-inf A and 2a-sup A (which are unique), where either A c_ 1-OD(X) or A _c (2, 1)-OD(X) although 2c-sup.4 exists even for .4 c 1 - O D ( X ) U (2, 1)-O~D(X) and A
N 1-O2)(X) - A1 7s 2~ 7s .42 -- A N (2, 1)-OT)(X).
Choose any subfamily ,4 c_ 1-OT~(X). Again applying Theorems 8.1 and 8.2 from [105], we find that there always exists sup.4 - V 1 A -- 7"1 int 7"1 el U A. If AEA
AEA
there exists a set B E (2, 1)-(.9Z)(X) such that A 4 B, that is, A c_ 7"1c l B for each A c .4, then 7"1 int 7"1cl U A c_ 7"1cl B so that sup A 4 B. Therefore sup A ACA
is l a - s u p A. Next, we shall consider a subfamily .4 c (2, 1)-OZ)(X) and show that
A2A-7"2int N A .~d
infA--
ACA
su A- V2A-7"2intT"lCl U
AC.A
AC.A
A
AEA
always exist. It is assumed that B c 1-OD(X) is a set such that B 4 A, that is, B C 7"1 el A for each A E A. We are to prove that B 4 inf.4 so that B c_ 7"1 cl 7"2 int n A. If we assume the contrary, that is, AEA
B n
(X \7-1C17"2 int N
.,
(X \7"2int N
then B N
AEA
AEA
But B 91-O~P(X), A E (2, 1)-OZ)(X) and, therefore, for each A c A the inclusion g C 7"1 clA implies 7"1 int 7"1 cl g C 7"2 int 7"1 cl A, that is, B c_ A for each A c A since T 1 C 7-2. Hence BC_ n
A and thus B C_ 7-lint n
AEA
Ac-7-2int n
ACA
which contradicts the inequality B O (X \ 7-2 int n 2a-inf A. Now we suppose that B c 1- 9 A E A. Then the obvious inclusion
A,
AEA
A) r z . Therefore inf A is
AEA
and A 4 B, that is, A c_ 7-1 cl B for each
s u p A - 7"2int T1 cl U A c_ 7-1 cl B AEA
implies that sup A is 2c-sup A. Finally, for a BS (X, T1 < 7"2) let us consider a subfamily A c_ 1-OD(X) u (2, 1)-O:D(X) such that .4 5 1-O2)(X) - A1 7~ ~ 7~ .42 - A n (2, 1)-OZ)(X). We shall show that the set 72 int T 1 cl (72 int T 1 clsup A1 U s u p A2) is 2a-sup A. For this it is sufficient to check that (1)-(3) of Definition 6.1.6 are fulfilled. (1) The inclusions sup A2 C_ 2a-sup A and sup A1 c_ T1 C12a-sup A are obvious.
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VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
Theorem 6.1.11. A goset (L, 4 ) is a G.lattice if and only if for any nonempty finite subset A c L, there exist i(:-inf A and/(;-sup A.
Proof. Clearly, it suffices to prove only the necessary part of the theorem. First we assume t h a t A c Li. If A = {x}, then the equalities i(j-inf A = i~.-sup A = x are obvious. Now let A = {x, y, z}. The proof will be carried out for infimum and s u p r e m u m simultaneously. By (1) of Definition 6.1.10, we have v = ic-inf(x,y ), t = i , - s u p ( x , y) and, therefore, there exist w = i , - i n f ( v , z), r = / ( ; - s u p ( t , z). Let us show t h a t w = i~;-inf A and r = / , - s u p A. We easily find t h a t w = inf A and r = sup A in the usual sense. Assume t h a t a c Lj is an element such t h a t a~
x, a ~
y and a ~
z
(x~
a, y ~
a and z ~
y
a, y 4
and v - i ( ~ - i n f ( x , y )
a).
Since a4
x, a 4
(x4
a)
(t-/(;-sup(x,y)),
by a) of (1) of Definition 6.1.10, a 4 v (t 4 a). By virtue of the same a r g u m e n t a4
v, a 4
z (t4
a, z 4
a)
imply a 4
w (r4
a).
This completes the proof for A - {x, y, z}. If A - { x 0 , x l , . . . , x n - 1 }, where n > 1, then the element i(:-inf((--, i(:-inf(i,-inf(xo, X l ),
X2),...,
Xn-1 )
( / ( : - s u p ( - . - i c : - s u p ( i , - s u p ( x o , X l ), X2 ), . . . , X n - 1 ) ) is i , - i n f A (/(:-sup A), which is proved by induction. Now we suppose t h a t
A N L ~ - A1 r 2~ r A2 - A N L 2 . Clearly, Ai is finite and, as in the first case, there exist x - j , - i n f Aj
(x - j(:-sup Aj ) and y - i~:-inf Ai ( y - / ( : - s u p Ai ).
Since (L, 4 ) is a G.lattice, by (2) of Definition 6.1.10, there exist v - i(j-inf(x, y) and t - i , - s u p ( x , y). Let us prove t h a t v - i , - i n f A and t - / , - s u p A in the sense of Definition 6.1.6. Indeed, by a) of (2) of Definition 6.1.10, v <_ y (y _< t), v 4 x i
i
(x 4 t) so t h a t v <_ inf Ai - i~;-inf Ai ( i(:-sup Ai - sup Ai _< t ) i
i
and v 4 inf Aj - j~;-inf Aj
(j(;-sup Aj - sup Aj 4 t).
Further, let
zcLi,
z4x
( x 4 z), z < y i
(y
T h e n by b) of (2) of Definition 6.1.10, z < v (t < z). Finally, if i
zcLj,
z
(x
i
and z 4
y ( y 4 z),
then by c) of (2) of Definition 6.1.10, z ~ v (t ~ z). Thus the elements v c Li and t E Lj satisfy all the conditions of Definition 6.1.6 so t h a t v - i , - i n f A (t - / ( ; - s u p A). D
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204
VI. Generalized Boolean Algebra and Related Problems
Then
(Xl Vj x2 V j . . . V j Xn) Vi (Yl ViY2 V i " " ViYn) -((x 1 Vj x 2 Vj " " Vj Xn_l) Vj xn) V i ((Yl Vi Y2 V i " " Vi Yn-1) Vi Yn) --
(x n Vj (Xl Vj x2 V j " ' V j
-
-
-
Xn_l) ) V i ((Yl Vi Y2 V i " " Vi Yn-1) Vi Yn) (Xn Vi Yn) Vi ((Xl Vj x2 V j " ' V j Xn_l) Vi (Yl Vi Y2 V i " " Vi Yn-1)) --
-
-
-
-
(Xn Vi Yn) Vi ((Xl Vi Yl) Vi (x2 Vi Y2) V i " " Vi (X,n-1 Vi Y'n-1)) -= (Xl Vi Yl) Vi (x2 Vi Y2) V i " " Vi (xn V~ Yn).
D e f i n i t i o n 6.1.13. Let s = {L = L1 U L2, A1, Vl, A2, V2} be a G.lattice and L~ c L~. Then s "- {L' - Ltl U L~, A1, V1, A2, V2} is said to be a G.sublattice of Z; if a c L~ U L~, b c L~ implies that a A~ b, a V~ b c L~. As we shall see in Sections 6.2 and 6.3, this notion is closely associated with the notions of G.ideal and G.filter. R e m a r k 6.1.14. Note t h a t by Definition 6.1.12, if an element x E Li is on the right-hand side of the operations Ai and Vi, then the definition of the domains of A~ and Vi necessarily implies that x c Li. This fact accounts for the absence of laws, corresponding to the commutativity laws in the well-known sense, t h a t is, the absence of the laws for Ai and Vi, corresponding to L2 when (x, y) c Lj x Li, though as we shall see in the sequel, such G.commutativity laws are available for G.lattices with the G.zero and G.unit elements. Our next important theorem shows the ways how we can pass from the G.lattice, defined by a goset to the G.lattice, defined by an algebra and vice versa. According to this theorem, no m a t t e r in which order we perform such pass-overs initially, they do not change the initial objects. Note that to prove the theorem, we do not need to assume that the G.lattice have the G.zero and G.unit elements. Theorem
6.1.15. The following conditions hold:
(1) Let a goset s = (L, 4 ) be a G.lattice. A s s u m e that x A~ y = i , - i n f ( x , y) and x Vi y = i~;-sup(x, y) for each pair (x, y) e (ni x ni) U (nj x Li). Then the algebra/~a = {L, A1, Vl, A2, V2} is a G.lattice. (2) Let an algebra s = {L, A1, V1, A2, V2} be a G.lattice. A s s u m e that y ~ x if and only if x Ai y = y for each pair (x, y) c (Li x Li) U (Lj x Li). Then 12q = (L, ~ ) is a goset which is a G.lattice. (3) If a goset s = (L, ~ ) is a G.lattice, then (s = s (4) If an algebra 12 = {L, AI~ Vx~ A2~ V2} i8 a G.lattice, then (s __ s
Proof. Conditions (1) and (2) will be proved first for i~-inf, that is, for Ai, and then f o r / , - s u p and so for Vi. Such a proof covers all the cases and we avoid going into a long tiresome discussion. (1) Let a goset Z; = (L, 4 ) be a G.lattice. By Definition 6.1.10 the binary operations Ai, Vi: (Li x L~)U (Lj x L~) ~ Li are defined as x Ai y = i , - i n f ( x , y) and x V~ y = ic~-sup(x , y). Let us show that the algebra s = {L, A1, V1, A2, V2}
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6.1. G o s e t s , G e n e r a l i z e d L a t t i c e s , . . .
207
Therefore, by b) of (2) of Definition 6.1.10, i , - s u p ( y , z) < i , - s u p ( j , - s u p ( x ,
y), z)
i
and thus i ( ; - s u p ( x , i ( ; - s u p ( y , z) ) <_ i G - s u p ( j ( ; - s u p ( x , y), z) i
so t h a t i.-sup(j.-sup(x,
y), z) - / , - s u p ( x , / . - s u p ( y ,
z)).
But this equality means t h a t (x Vj y) V~ z - x V~ (y V~ z). In a similar manner, using Definition 6.1.10, we can prove t h a t (xAjy) A~z--xA~(yAiz)
for x, y C L j ,
z C L~,
or x, z c L~,
y c Lj.
T h u s condition GL3 of Definition 6.1.12 is satisfied. If x c Lj, y c L~, t h e n by a) of (2) of Definition 6.1.10, i(;-inf(x,y) 4 x and, therefore, b) of (2) of the same definition implies j ( x - s u p ( i ( ~ - i n f ( x , y ) , x ) < x. J T h u s j ( j - s u p ( i ( j - i n f ( x , y ) , x ) - x since the inequality x <_ j ( ~ - s u p ( i ( ~ - i n f ( x , y ) , x ) J is also obtained by a) of (2) of Definition 6.1.10 so t h a t (x A~ y) Vj x -- x. F u r t h e r m o r e , by a) of (2) of Definition 6.1.10, y <_ i , - s u p ( x , y ) and, therefore, i
y so t h a t (x Vi y) A~ y -- y. In a similar manner, using
i~:-inf(i(~-sup(x, y), y) -
Definition 6.1.10, we (:an prove the equalities (x V~ y) Aj x - x and (x Ai y) V~ y -- y. Thus condition GL4 is also satisfied and s _ {L, A1, V1, A2, V2} is a G.lattice. (2) Let the algebra s - {L, A1, Vl, A2, V2} be a G.lattice. Assume t h a t y4 Then 4
x .z--->xAiy--y
for each pair (x,y) c ( L i x L i )
U(Lj xLi).
L, --_< are partial orders on Li. Moreover, the binary relation 4
is a
quasi order on L - L 1 U L 2 . Indeed, it is obvious t h a t x 4 x for each x c L. Let us show t h a t x 4 Y and y 4 z imply x 4 z for each elements x, y, z c L. To this end, we shall consider the cases: x, y, z 6 Li;
x 6 Li,
y, z C L j ;
x, y C Li,
z 6 Lj;
and x, z c Li,
T h e case x, y, z c Li is obvious since 4 IL, =<--. i
If x c L~, y, z c L j , t h e n x 4 y ,,+----5, y A ~ x -- x and y 4 z ,z-----~, z A j y -- y.
Therefore x - y Ai x -- (z
t h a t is, x - - z A ~ x ~ x 4 If x, y E L~, z E L j , then
Aj y) Ai x and by GL3, we have z.
x 4 y ,z-~, y A ~ x -- x and y 4 z ,,v----5, z A ~ y -- y.
Therefore x - y Ai x -- (z A~ y) Ai x and by GL3, we have (zAiy) Aix--zAi(yAix)--zAix, t h a t is, x - - z A i x ~ x ~
z.
y c Lj.
208
VI. Generalized Boolean Algebra and Related Problems
Finally, we assume t h a t x, z c Li, y c L j . T h e n x4
y ,,r
y Ai x -- x,
Y4
z ,~-~, z Aj y -- y.
Therefore x - y At x -- (z Ay y) At x and by GL3, we obtain (Z A j y ) A i X - - z Ai
(y At x) - z Ai x
so t h a t x - z At x ~ x 4 z. T h u s the relation 4 is a quasi order on L - L1U L2 such t h a t 4 IL, =_< and s o / : q - (L, 4 ) is a goset. i
Now, assuming t h a t i a - i n f ( x , y) - x A~ y and i(~-sup(x, y) - x Vi y for each pair (x, y) c (Li x L i ) U ( L j x L i ) , let us prove t h a t the g o s e t / : q - (L, 4 ) is a G.lattice in the sense of Definition 6.1.10. First, we show t h a t if x c L j and y E L i , then i~-inf(x, y) - y <--5, i t - s u p ( y , x) - x, t h a t is x Ai y -- y <--> y Vj x -- x. Indeed, by GL4, we have x A i y - - y implies y V j x implies x Ai y -- (y V j X) A i y - - y. Therefore Y4
(xAiy)Vjx-
x and y V j x -
x
x <----~, x A i y = y ,,+--~, y V j x = x.
Now we proceed to considering the fulfilment of (1) and (2) of Definition 6.1.10. Let (x, y) E Li x Li be any pair. Then, according to the a r g u m e n t s t h a t follow Definition 6.1.12, i , - i n f ( x , y) a n d / , - s u p ( x , y) coincide with inf{x, y} and sup{x, y}, respectively, in the usual sense. Moreover, if z c L j and z 4 x, z 4 y, t h e n by GL3, (xA~y) Ajz--xAj(yAjz)--z
and so z 4
xAty~z4
inf{x,y}.
Quite in a similar manner, one can prove t h a t x ~ z Ai y 4 z imply sup{x, y} 4 z. Hence a) is also satisfied and thus (1) of Definition 6.1.10 is completely valid. If ( x , y ) c L j x L~, then by GL4, (x A~ y) V~ y -- y so t h a t x At y _< y and, i
therefore, i~.-inf(x, y) <_ y. Similarly, (x A~ y) Vj x -- x so t h a t x A~ y 4 x, t h a t is, i
i , - i n f ( x , y) 4 x. F u r t h e r m o r e , let z E Li, z _< y and z 4 x. Then, by GL3 and i the definition of 4 , we have z -
y
-
y -
(x
z)
y -
x
A, y)
-
(y
-
y)
so t h a t z < x Ai y and, therefore, z < i~-inf(x, y). Finally, let z E L j and z 4 y, i
z < x. Then, by GL3, J
i
(X A i y ) A j z -- x A j ( y A j z) z x A j z z Z a n d s o
z ~
x
At y,
t h a t is, z ~ i~j-inf(x, y). Using a similar reasoning and taking into account the equivalence y ~ x <---> y Vj x = x, we can prove the fulfilment of the rest of the conditions for the s u p r e m u m case in (2) of Definition 6.1.6. Thus the goset L;q = (L, ~ ) is a G.lattice. (3) T h e gosets L; = (L, ~ ) and (L;a) q have the same basic set L = L1 U L2. Hence to prove t h a t L; = (L;a) q, it suffices only to show t h a t the quasi orders on L; and (L;a) q coincide. First, we shall prove t h a t if the goset/2 = (L, ~ ) is a G.lattice, then y ~ x ~ i c - i n f ( x , y ) = y for each pair (x,y) c (Li x L i ) U ( L j x L~).
6.1. Gosets, Generalized Lattices, ...
209
Indeed, if (z, y) c Li x Li, this equivalence is obvious since ~ IL, ----<. Now, let i
(z, Y) C L j x Li and 9 ~ z. By a) of (2) of Definition 6.1.10, i , - i n f ( z , y) <_ 9 since /
s - (L, 4 ) is a G.lattice. On the other hand, y _< y and y 4 x. Hence, by b) of i
(2) of Definition 6.1.10, y < iG-inf(z, Y) so that y i
i~-inf(z, y).
Conversely, if the goset s - (L, 4 ) is a G.lattice, then by a) of (2) of Definition 6.1.10, y 4 z. Thus, if a goset s - (L, ~ ) is a G.lattice, then y 4 z implies i~;-inf(z, 9) - Y for each pair (z, 9) c (Li x L i ) U ( L j x Li). Therefore, to prove that (s _ s it remains to apply successively (1) and (2) of Theorem 6.1.15. (4) The algebras s - {L, A1, VI,A2, V2} and (s have the same basic set L -- L 1U L 2. Hence to prove that (s _ s it suffices to show that the operations on s and (s coincide by applying successively (2) and (1) of Theorem 6.1.15 since s {L, A1, V l , A2, V2} is a G.lattice. K] Based on Theorem 6.1.15, we denote by s - { L 1 , A 1 , V l , 4 , L 2 , A2, V2} a G.lattice, defined by means of a goset. It is obvious that s - {L1 - {(9, e},A1,VI, d , L 2 -- {19, e},A2, V2}, where A1 -- A2 -- A and V1 - V2 - V are usual lattice operations, is a G.lattice and s is a G.sublattice of any G.lattice t; - {L1, A1, V1, 4 ,L2, A2, V2} such that L1 n L2 - {19, e}. In the sequel we shall consider only G.lattices with the G.zero and G.unit elements, that is, of the type t ; - {L1,A1,V1,19,4 ,e, L2, A2, V2}, where 19, e c L1K1L2.
Our next theorem is most important for further consideration because it shows that the posets, composing a goset, are in a one-to-one correspondence of special type. Theorem
6.1.16. Let s - {L1, A1, Vl, 19, 4 , e, L2, A2, V2} be a G.lattice and ()~,,X~) be a pair such that X, " Li ~ L j are maps, defined as follows" X, (x) - x Vj 19 f o r each x c L~. T h e n the conditions below are satisfied: -
(1)
(x)
-
9
a ,d
(e)
-
e,
-
(2) X, o X, -- idL, so that X, (X, (x)) - x f o r each x c Li; thus X1, X~_ are bijections and X, - X T, 1 (3) X, o Ai -- A j o (X,, X,) a n d X, o V~ - V j o ( X , , X , ) so t h a t
and T h u s x <_ y ~ }(, (x) <_ X, (Y) f o r each pair (x, y) c L~ x L~ i j (the i s o t o n i c i t y of the m a p s X,). (4) }(, o Ai -- Aj o (~,,,}(,) and X, o V~ - Vj o ( X , , X , ) so that
and
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y) -
v ; x, (y).
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212
VI. Generalized Boolean Algebra and Related Problems
()~1,)~2)
The pair X -
is called the G.identity operator.
c
b
x. ~. x. .~ . I j -~
d
Let (L = L1 U L2, 4) be the goset in Diagram 2, where the quasi order relation x ~ y is determined as for Diagram 1 by the arrangement of elements x, y E L 1 U L 2 = { a , b , c , n } U { c , d , m , n } at different or the same levels.
n
Diagram 2 If we define the binary operations Ak, Vk : Li x x 4 y <--5, x V k y = y for each pair ( x , y ) c verification of axioms L1 - GL4 shows t h a t s = G.lattice with (9 = n, e = c, X1 (a) = m, ~1 (b) --- d ,
Lk ~ Lk as y A~ x = x ,', ;, Li x Lk, then an elementary { L 1 , A 1 , V1, 4 , L 2 , A2, V2} is a
X2 (~'~) :
a
and X2 (d) = b.
P r o p o s i t i o n 6.1.17. For a G.lattice s = {L1, A1, V1, 0, 4 , e , L2, A2, V2}, the following conditions are satisfied:
(1)
(x /~ y) v~ (x /~ z) = x / ~ (y v~ (~ /~ ~)) ., ,.. .'. :, (z ~ x implies (x Ai y) Vi z -- x Ai (y Vi z)) z. ;.
.'. :. (x v~ y) A~ (x v~ z) - x v~ (y A~ (x v~ z)) ,'. ;. ,'
:. (x ~ z implies (x Vi y) Ai z -- x Vi (y Ai z))
if x, y, z E Li or x C L j , y, z c Li.
(2)
(x Aj y) v~ (~ A~ z) = x A~ (y v~ (~ A~ ~)) ,, ,, .',
,~
(z ~ x implies (x Aj y) Vi z = x Ai (y Vi z))
.'
:.
.' :. (x vj y) A~ (x v, z) = x v~ (y A~ (x v~ z)) .'. '. -(
:.
(x 4 z implies (x Vj y) Ai z = x Vi (y Ai z))
if x, y E L j , z 6 Li or x, z E L i , y E Lj. Proof. Since all these conditions are proved in a similar manner, we can do with proving only (2) for x, y c L j , z 6 Li. First, let (xAjy) Vi(xAiz)--xAi(yVi(xAiz))
and z 4
x.
T h e n z = x A~ z and hence
(x Aj y) v~ ~ = (x A~ y) v~ (x A~ z) = x A~ (y v~ (~ A~ ~)) = ~ A~ (y V~ ~). Conversely, let z ~ x imply (x Aj y) V~ z = x A~ (y V~ z). Then x A~ z ~ x implies that
(x Aj y) v~ (x A~ z) = x A~ (y v~ (x A~ z)) and thus the first equivalence is true.
6.1. Gosets, Generalized Lattices, . . .
213
Using a similar reasoning one can prove that the trird equivalence is also true. By virtue of GL3 and (5) of Theorem 6.1.16, it suffices to prove only that (z 4 x implies (x Aj y) Vi z -- x Ai (y Vi z)) ~' ,<---5, (x 4 z implies (x Vj y) Ai z -- x Vi (y Ai z)). If x 4
z, t h e n x V i z - z a n d
(x v j y) A~ ~ - (x v j y ) / ~ (x v~ ~) - ((x v~ ~) nj (y v5 x)) A~ ~ -
= (((~ v~ ~) Aj y) v~ ~) v~ e - ((z Aj y) v~ ~) v~ e -- (x v j ( z / ~ y)) v~ e - x v~ ((z/~j y) v~ e) - x v~ ( y / ~ ~). Conversely, z ~ x implies x Ai z -- z and
(x A~ y) v~ z - (x Aj y) v~ (x A~ ~) - ((x A~ ~) v~ (y Aj ~)) v~ e -
= (((x A~ ~) vj y) Aj x) v~ e - ((z vj y) A~ x) v~ e - (x Aj (~ vj y)) A~ ~ = x / ~ ((~ vj y) n~ ~) - x / ~ (y vj z).
n
D e f i n i t i o n 6.1.18. A G.lattice s - {L1, A1, Vl, O, 4 , e, L2, A2, V2} is said to be modular (or Dedekind) if it satisfies one of the equivalent conditions both in (1) and (2) of Proposition 6.1.17. This notion is closely associated with D e f i n i t i o n 6.1.19. A G.lattice t; - {L1, butive if the following laws hold:
A1, V l , I~, 4 , e,
L2, A2, V2} is distri-
CDL1 x A~ (y V~ z) -- (x A~ y) V~ (x A~ z) and x Vi (y Ai z) -- (x Vi y) Ai (x Vi z) ifx, y, z c L i o r x 6 L j ,
y, z c L i .
x v~ (y A~ z) - (x vj y)A~ (x V~ ~ ) i f x, y C f j , ~ c f~ or x, z c f~, y c f j . G u n 2 (x v~ V) n~ z - (z A~ z) v~ (v A~ z) and (xAiy) Viz--(xViz)Ai(yViz) ifx, y, z E L i o r x E L j , y, z c L i . (xVjy) A~z--(xAiz)Vi(yAiz) and ( x A j y ) Vi z -- (x Vi z) A~ (y Vi z) if x, y C L j , z ~ L i o r x , z~Li, y~Lj. It is obvious that if s -- {L1,A1,V1,Q),4 ,e, L2, A~,V2} is a distributive G.lattice, then {Li, Ai, Vi, (9, e} are distributive lattices in the usual sense and every G.chain is a distributive G.lattice. GDL1 and GDL2 are called the G.distributivity laws. Let us denote by I GDL1, II GDL1, III GDL1, and IV GDL1 the first, second, third, and fourth equalities in GDL1, respectively, and similarly for GDL2. Theorem have:
6.1.20.
For a G.lattice s
IGDL1 ~
IIGDL2,
-
{L1,A1,VI,~,4 ,e, L2, A~,V~}, we
IIGDL1 ~
IGDL2
and
IIIGDL1 <---->, IVGDL2, so that GDL1 ~
IVGDL1 ~
IIIGDL2
GDL2.
Proof. All the proofs essentially utilize Definition 6.1.12 and Theorem 6.1.16.
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220
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
= y Ai ((a Aj x) Vi (x Ai y)) -- (a Aj x) V~ (x Ai y) --
-- (a Aj x) Vi (a Ai y) Vi (x Ai y) and so x = y Vj @. As for the implication (4) ~ (3), it is an immediate consequence of the notion of a G.neutral element and Theorem 6.1.22. D C o r o l l a r y 6.1.25. For a G.lattice 12 = {L1,A1,V1,Q),4 ,e, L2, A2, V2} the following conditions are satisfied: (1) Every G.neutral element is G.standard and coG.standard. (2) Every G.standard (coG.standard) element is (V,A)-distributive ((A, V)-distributive). (3) Every G.standard and c o G . s t a n d a r d or every G.standard and (A, V)-distributive element is G.neutral.
(1) By (3) and (4) of Theorem 6.1.24, every G.neutral element is (V, A)-distributive, (A, V)-distributive, and
Proof.
aA~x=aA~y,
aV~x=aV~y
aAjx=(aAiy)
Aje,
imply x = y
aVjx=(aViy)
for x, y c L ~ ,
V j O imply x = y V j @
for x C Lj, y E L~. Therefore it remains to use (1) .z---->.(2) in Theorem 6.1.24. Assertion (2) follows directly from the equivalence (1) .z----5. (2) in Theorem 6.1.24. (3) If an element a e Li is G.standard and coG.standard, then by (1) <---5, (2) in Theorem 6.1.24, it is (V, A)-distributive, (A, V)-distributive and a Ai x = a Ai y, a Vi x = a Vi y imply x = y for x, y C Li,
aAjx=(aAiy)
Aje,
aVjx=(aViy)
V j O imply x = y V j O
for x E L j , y E L~. Thus it remains to use (3) z---> (4) in Theorem 6.1.24. If an element a c L~ is G.standard and (A, V)-distributive, then by (1) ~ in Theorem 6.1.24, it is (V, A)-distributive and a Ai
X --- a A i y ,
aAjx=(aAiy)
a Vi X
Aje,
=
a Vi y
(2)
imply x = y for x, y c Li,
(a Vj x) = (a Vi y) Vj (9 imply x = y V j ( 9
for x C Lj, y E Li. Thus it remains to use (3) ,z---->,(4) in Theorem 6.1.24.
[3
D e f i n i t i o n 6.1.26. A G.lattice s = {L1, A1, V1, 0, ~ , e , L2, A2, V2} is said to be G.complemented if there exists a pair ~ = (pl, p2) such that Pi : Li ---, Lj are maps and x Aj ~ ( x ) = @, x Vj ~ ( x ) = e for each element x c L~. The pair p = (~1, ~2) is called a G.complementation operator. P r o p o s i t i o n 6.1.27. For a distributive G.lattice s = {L1, A1, V1, 0, 4 ,e, L2, A2, V2} the G.complernentation is unique.
6.1. Gosets, Generalized Lattices, ...
221
Proof. Let q p ' - (p~, p~) be another pair, where ~{'L~ -+ Lj are maps such that x Aj p~(x) -- O and x Vj p~(x) - e. Then by GDL2 we obtain ~ ( x ) - e v~ ~ ( x ) - (. Aj ~'~(.)) vo ~ ( . ) = (x vj ~ ( . ) )
Aj ( ~ ( ~ ) vj ~ ( . ) )
- ~ Aj (~'~(.) vj ~ ( . ) )
- ~'~(.) vj ~(~)
so that ~{(x) _< ~i(x). The case ~i(x) <_ ~{(x) is proved in a similar manner and thus ~ - ~ .
j
J
D
Now we are ready to introduce the basic notion. D e f i n i t i o n 6.1.28. A G.Boolean algebra (briefly, GBA) is a distributive and G.complemented G.lattice. In the sequel a GBA will be denoted by A - {A1,A1, V I , ~ I , O , ~ , e , A2, A2, V2, ~2}. It is obvious that if A1 - A2 - A, then A1 -- /~2 -- A and V1 -- V2 -- V are usual lattice operations, (~1 - - ( P 2 - - - - is & complementation operation in the usual sense and, therefore, . 4 - {A, A, V , - , O, e} is a Boolean algebra (briefly, BA) in the usual sense. A G.Boolean subalgebra (briefly, GBSA) of a GBA A is a set B c_ A1 U A2, B • A1 - B1 ~ Z ~ B2 - B A A2, which is closed under the four G.Boolean operations: Ai and ~ , or, V~ and pi. It is obvious that any GBSA contains 0 and e. Take place the following obvious statement.
Proposition
6.1.29.
Yo~ ~ y
~o~-~.~pty fa.~ily {B~- B~ u B ~ } ~ of
GBSA's of a GBA A, the intersection Bo -
( ~ B~) U ( ~ Bt2) is a GBSA tET
o/A.
tET
Therefore for any subset D c A1 U A2, there exists a GBSA of At generated by D, which is the smallest GBSA containing D. Clearly, for a GBA A - { A 1 , A 1 , V I , ( ~ I , ~ , 4 ,e, A2, A2, V 2 , ~ 2 } , the subset {a, ~ l ( a ) , ~=},e} U {X1 (a), (~l(a), {~, e} C n l U n 2 together with the corresponding G.lattice operations is a GBSA. E x a m p l e 6.1.30. Let us consider a nonempty BS (X, the binary operations
A~, v~. ( ( ~ , j ) - o ~ ( x ) •
T1
~S
7"2)
and define
( i , j ) - o ~ ( x ) ) ~ ((j,i)-o~(x)• ( i , j ) - o ~ ( x ) ) -~ -~ (~, j ) - o ~ ( x )
as follows" U A~ V -
T~int Tj cl U ~ V and U V~ V - ~-~int rj cl(~-~ int T1 cl U U V).
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6.1. Gosets, Generalized Lattices, . . .
225
Now we assume that (x, y) ~ A j x Ai and x ~ y. Then, by Theorem 6.1.15, L2 and GL3, we have x
J(;
~ =
9no ~ ( ~ ) = (~ n , x) n , ~ ( ~ ) = ~ ( ~ ) n , (~ n , ~) = - ( ~ ( ~ ) n~ ~) no ~ = ~.
On the other hand, by (1) above and (3) of Theorem 6.1.31, we have x - y = (9 <----> x Aj ~ i ( y ) = (9 ~ Jc;
q~j(x) Vi y = e
and hence ( p j ( x ) V ~ y ) A j x = x so that by GDL2, ( ~ j ( x ) A j x ) V j ( y A j x ) = y A j x = x. Thus x 4 y. (6) We shall only consider the conditions without brackets. By condition (3) above
(x,v) e p~(~,) ~
v _< ~,(x) ~ z
< :, x < ~ i ( y ) ~ J
~ ( ~ , ( . ) ) _< ~(v).: j
( y , x ) ~ PA(q~i).
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y <_ ~o(x) ~ i
~o(~(y)) _< ~ ( x ) . :
:.
i
(~(v), ~ , ( , ) ) e P~(~,) and, finally, i
j
< ', (~j(x), ~ ( y ) ) 9Pv(~i). (7) Let us prove, as an example, the condition in the brackets. If (x, y), (z, v) c Pv(Fj), then p j ( x ) <_ y and p j ( z ) < v. Therefore qpj(x) V, p j ( z ) <_ y V~ v and, i
by (2) of Theorem 6.1.31,
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i
q~j(X Aj z) <_ y Vi v, that is, (x A j z , y V i v ) C P v ( q ~ j ) -
On the other hand, F j ( x ) < y, F j ( z ) <_ v imply F j ( x ) Ai qpj(z) < y Ai v i i i
and thus p j ( x Vj z) <_ y Ai v gives (x Vj z, y Ai v) C Pv(~j). i
K]
Theorems 6.1.16 and 6.1.31 give rise to some interesting statements related not only to generalized, but also to ordinary algebraic structures. C o r o l l a r y 6.1.33. Let A = {A1, A1, V1,991, (~, ~ , e , A2, A2, V2, p2} be a GBA and ~ = (~1, ~2) be a pair such that ~ : Ai ~ A~ are m a p s defined as follows: ga~ = ~ j o Xi so that ~b~(x) = ~ j ( X , ( x ) ) f o r each x r A i . T h e n the following conditions are satisfied: (1) ~ = x, o ~ ~o that ~ ( x ) = X, ( ~ ( ~ ) ) fo~ ~ach 9 eA~. (2) ~&(O) = e and ~b,(e) = 0 . (3) ~b~ o ga~ = idA, so that ~b,(~b~(x)) = x f o r each x c Ai.
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6.1. Gosets, Generalized Lattices, . . .
229
(8) By analogy with the proof of (6) of Corollary 6.1.32, for each pair (x, y) c
Aj x Aj, we have j
J
Note now t h a t ~j o @j - ~i o ~j. Indeed, ~j
O ~)j - - ~ j o ~ i O Xj - - Xj
and ~i o q~j - ~i o ~i o Xj - ;~j.
Moreover since (x, y) e PA(~j) <---> Y <_ ~j(x) and ~j are antitone, we have J
~ j ( ~ j ( ~ ) ) <_ ~j(y) ~
~(~j(~))
i
_< ~j(y) ~
(~j(x), ~j(y)) e P ~ ( ~ ) .
i
Therefore
(~j(x), ~ j ( y ) ) e Finally, (x, y) e PA (~j) ~
P~(~)~
(~j(y), ~ j ( x ) )
9P ~ ( ~ ) .
Y < ~j (x) and, by (4) above, J J
(9) We will prove only four implications; the others can be proved similarly. I f ( x , y ) e PA(~j), z 9Li and z 4 y, then by (7) above, y ~ ~j(x) s o t h a t z < ~ j ( x ) s i n c e ~ A~ --<. Therefore (x,z) 9PA(~j).
j
Now, if (x,y)
J 9P v ( ~ j ) , z
9n~ and x 4
z, then we have ~j(x) < y and J
x 4 z. But ~j(x) < y ~ ~ ( y ) < x so t h a t ~ ( y ) ~ z and, by (7) above, we i j have (z,y) 9Pv(~i). The rest can be proved in a similar manner. Assertion (10) is proved in the same way as (7) of Corollary 6.1.32. (11) By (7) above, for each x 9A~, the element ~ ( x ) 9A~ is its complement in the usual sense and so ~i(x) is unique. Hence Xl
-
{A1,
A1, V l , ~)1, O , S , 1
e} and ,.,42 -- {A2, A2, V2, ~2, (~, <_, e} 2
are BA's. To prove t h a t ~, "Ai ~ Aj are isomorphisms, by virtue of (2) and (3) of Theorem 6.1.16, it suffices to prove only t h a t )/, (~i(x)) - ~j(X, (x)) for each x 9Ai. By (5) above ~ ( x ~ (x)) - ~ j ( x v~ e ) - ~ ( x ) / ~
~(e)
- ~(x)/~
~ - x~ ( ~ ( x ) ) .
Furthermore, let a 9Ai be an atom of Ai and X~ (a) 9Aj be not an atom of My. Then there exists an element b 9Aj \ {O} such that b < X, (a) - a Vj (9. We J
have ;~ (b) < ~ ()~ (a)) - a, where X;~(b) # (9 since X~ are isotonic bijections. The contradiction obtained shows t h a t )/, (a) is an atom of Aj. The reverse implication is obvious since )/j ()/~ (a) - a. Proving the second equivalence is clear. (12) First, let (x,y) 9A~ x A~. Then x 4 y ~ x < y <---5. x A ~ y - x. Therefore
x - ~ - x a~ ~ ( ~ ) - (x a~ ~) a~ e~(~) - x a~ (~ a~ e~(~)) - x a~ e - o. i
230
VI. G e n e r a l i z e d Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
Conversely, if x - y i
Oi(xAiOi(y))-
- x/~i ~i(Y) - 0 , t h e n by (2), (3), a n d (4) above, we have
Oi(x)Viy-
e a n d so, by ( 7 ) a b o v e , ( r
Now let (x, y) E Aj x Ai. T h e n x 4 y ~ x - y -
r
i
-
x < y.
y Aj x -- x. T h e r e f o r e
(y Aj
= ((r
y) e Pv(r
-
A, y) Aj
(r
A,
Aj (y Aj x ) ) A,
-
- e.
O n t h e o t h e r hand, if x - y - O, t h e n e i ( x Ai ~i(Y)) -- e j ( x ) Vi y -- e. T h e r e f o r e i
(~j(x), y) c P v ( ~ j ) a n d hence p j ( ~ j ( x ) ) <_ y, t h a t is )/j (x) - x Vi O _< y z----> x 4 y. i
i
(13) We will prove only t h e equivalence in brackets. i
g~i(Y) <_ x Vi (9 ~ ( y , x Vi (9) C P v ( e i ) . i I m m e d i a t e c o n s e q u e n c e of previous r e a s o n i n g is ": :" ~i(Y) 4 x Vi (9 ~
Example
6.1.34.
I-I
C o n t i n u i n g E x a m p l e 6.1.30, let us consider t h e pairs X First, let U e ( 1 , 2 ) - O Z ) ( X ) - 1-OT)(X). T h e n by T h e o r e m 6.1.16 a n d (1) of L e m m a 0.2.1, we have
(X1,X2) a n d ~ - ( ~ 1 , ~ 2 ) .
X1 (U) - U V2 2~ - 7-2 int 7-1 cl (7-2 int 7-1 cl U U 2~) - 7-2 int 7-1 cl U. 9(2, 1)-O~P(X) - 2-OZ)(X), t h e n by (3) of C o r o l l a r y 2.1.7, we have
Now, if U
)/2 (U) - U V1 2~ - 7-1 int 7-2 cl (7-1 int 7-1 cl U U 2~) = 7-1 int 7-2 cl 7-1 int 7-1 cl U - 7-1 int 7-2 cl U. Hence ( 1 ) o f L e m m a 0.2.1 a n d ( 3 ) o f C o r o l l a r y 2.1.7 give )(~2(~1 ( U ) ) -- T2 int --
Wl
T1
cl U
V1 ~
-- T1
int r2
(T 1 int 7-1 cl r2 int
cl
T1
cl U U 2~) -
int r2 el rx int 7-1 el r2 int 71 cl U - 7-1 int r2 cl r2 int 71 cl 72 int r l el U -
-- 71 int r2 cl 72 int 7-1 cl U - 71 int r2 cl 71 int 7-1 cl U - 71 int r2 cl U - U so t h a t X~ (XI ( U ) ) )(~1
()(~2(U))
T1
--
U and int r2 cl U V2 2~ - rg, int r l cl (r2 int
= r2 int r2 cl r2 int r2 cl
T1
T1 C1 T1
int r2 cl U U 2~) -
int r2 cl U - r2 int r2 cl r l i n t r2 cl U =
= r2 int r2 cl 72 int r2 cl U - U, a n d so X I ( X 2 ( U ) ) @I(U)-
U. F u r t h e r m o r e , if U
X2(~I(U))-
~2(T2
9(1, 2 ) - ( 9 / ) ( X ) -
i n t ( X \ U))
-T1
1-O~D(X), t h e n
int 7-2 c172 i n t ( X \ U) -
= r l int r2 cl r2 int rl c l ( X \ U) and, by (3) of C o r o l l a r y 2.1.7, we o b t a i n ~)1 ( g )
- - T1
int 7-2 el T1 int - - 7"1
T1
c l ( X \ U)
- - T1
int ~-2 cl
T1
int ~-2 c l ( X \ U) -
int 7"2 c l ( X \ U) - 7"1 i n t ( X \ U).
6.1. Gosets, Generalized Lattices, . . .
231
Consequently, g A 1 ~)1 ( g )
-
T 1 int r2 cl U ~ T 1 i n t ( X \ U) -
U N T 1 i n t ( X \ U) - 2~,
U Me ~)1 ( g )
-
T 1 int 72 cl (T1 int T1 C1U [_J T1 i n t ( X \ U)) -
= 71 int (72 cl 71 int T 1 cl U U 72 cl 71 i n t ( X \ U)) = 71 int (72 cl U U 72 cl 72 i n t ( X \ U)) = 71 int (72 cl U U 72 cl(X \ 72 cl U)) - X. Now, if U c (2, 1)-OT)(X) - 2-OT)(X), t h e n ~2(U) - X1 (~2(U)) - X1 (71 i n t ( X \ U)) - 72 int 71 cl 71 i n t ( X \ U) = 72 int 71 cl 71 int 72 cl(X \ U) - 72 int 72 cl 72 int 72 cl(X \ U) = 72 int 72 cl(X \ U) - 72 i n t ( X \ U). Therefore U A2 ~2(U) - 72 int 71 C1 g A 7"2 i n t ( X \ U) - U N 72 i n t ( X \ U) - 2~, U V2 ~2(U) - 72 int 7-1 cl (72 int T 1 cl U U 72 i n t ( X \ U)) = 72 int (71 cl ~-2 int 7-1 cl U U T 1 cl 72 i n t ( X \ U)) = 72 int (71 cl U U T 1 c l ( X \ 72 cl U ) ) -
X.
Finally, ~i(~i(U))
- 7i int ( X \ 7i i n t ( X \ U ) ) - 7i int 7~ cl U - U
for each U c ( i , j ) - O D ( X ) -
a
i-07P(X).
c
Let ( A = {A = A l U A 2 = {a, b, c, d, m , n}, 4 } ) b e the goset in D i a g r a m 3 where A1 = { a , b , d , m}, A2 = { b , c , d , n}, and the quasi order relation on A is defined as follows: if x, y c Ai and x is connected with y by the solid line directed upwards from x to y, then x -< y; if x and y are connected by the horizontal broken line, t h e n x -< y and y -< x. Finally, i f x , y c A \ { b , d } are not connected, t h e n x and y are not comparable.
Diagram 3 Let binary operations Ak, V k : A i x Ak ~ Ak be defined as follows: y A i x = x ,z-----+, x ~ y ,z-----~, x V k y = y
for each pair of c o m p a r a b l e elements (x, y) c Ai x Ak and yAix=xAky=d,
yVix=xVky=b
for each pair of n o n - c o m p a r a b l e elements (x, y) c Ai x Ak. T h e n it is not difficult to see t h a t A = { A 1 , A 1 , V l , ~ 1 , 0 , 4 , e , Ae, A2, V 2 , ~ 2 } is a G B A , where (3 = d, e = b, ~1 ( a ) ~-~ ?~, ~1 (T/~) - - C, ~ 2 ( C ) - - T/~, ~2(Tb) = a , ~ ) l ( a ) = ?Tt,
e l ( ? T t ) -- a ,
r
= /t,
e2(/t)
~-- c,
232
vI. Generalized Boolean Algebra and Related Problems
Xl(a)-
c,
xl(m)-n,
x2(n)-m.
X2(c)-a,
Next, for a GBA A - {A1, A1, VI,~I, Q), ~ ,e, A2, A2, V2, ~2}, we shall define two more binary operations >" Ak x Ai ~ Ai in the manner as follows" iG
x-->y-~k(x)
Viy if ( x , y ) ~ A a x A i .
T h e o r e m 6.1.35. For a GBA .4 = { A 1 , A I , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2,~2}, the following statements hold: (1) ~i(x ~
y) = x - y if (x,y) c Ak x Ai.
z(;
(2) ~ - - .
iG
x
iG
JG
x = ~, e - - - .
x = ~, ~ - - .
i(;,
y) ~
(x ~
y) ~
z~
x = x and
kG
ZG
~G
y = x V~ y if (x,y) c Ak x Ai.
(4) x ~ y = e ~ x ~
y/f(x,y)~AkxA~.
zG
Zc;
x
i(;
e = ~(~)
iG
>e=eifxEAi.
(3) (x ~
(5) x ~
9 =x, ~ - . .
i~
y = (~(y) ~
~ ( x ) ) V~ O if (x, y) e A~ x A~ and
3c;
> Y = W(Y) ~
V j ( x ) if ( x , y ) E Aj x A~.
zc;
(6) x /~ (x ~
y) = x / ~ y, y / ~ (x ~
ZG
y) = y and
~G
x ~
(x Ai y) = x ~
y if (x,y) c Ak x Ai.
y ~
(x A~ y) = y ~
x if (x,y) c A~ x A~ and
y ~
(x Ai y) = (y ~
iG
zG
(7) x ~
3~
x ~
iG
3~
~ ( x ) if (x,y) c A~ x A~ and
~(y) = y ~
3~
~i(Y) = (Y ~
JG
x) Vi (3 if (x,y) E Aj x Ai.
V j ( x ) ) Vj (3 if (x, y) c A j x Ai.
z c;
(s)
(x ~
y)/~
zo
(x ~
z) = x
---., (~/~ ~)
zG
iG
if x, y, z ~ A~ or x ~ Aj, y, z ~ A~ and
(~ ~
~) ~ (x ~
3c;
z) = ~ ~
ZG
(~ ~ ~)
iG
if x, y c A j , z c Ai or x, z ~ Ai, y ~ A j . (9)
(x
~
~)/~ (~ ~
zo
z) = (~ v~
~) ---.
zG
z
iG
if x, y, z ~ A~ or x ~ Aj, y, z ~ Ai and
(x ~ zG
z)
~ (~ ~
z) = (x v~ ~) ~
zG
z
iG
if x, y ~ A j , z ~ Ai or x, z ~ Ai, y ~ A j .
(10)
x --.~ (y ~ iG
zG
z) = (x Ai y) ---> z = y ~ iG
~G
(x ~
zG
z)
a~a
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6.1. G o s e t s , G e n e r a l i z e d
Lattices, ...
237
Once again to emphasize the importance of the G.identity operator, recall that to define a GBA above, we first introduced the notion of a goset, then successively the notions of a G.lattice, a distributive G.lattice, a G.complementation operator and, finally, the notion of a GBA. Now that we have fully covered the (fundamental) details, we can introduce the notion of a GBA more easily by using only the well-known notions of the theory of Boolean algebras and the G.identity operator. T h e o r e m 6.1.37. Let ( L - L1 U L2, 4 ) be a goset, where (L~, A~, V~, (9, e) are lattices and L1 UI L2 - {0, e}. If there exists a pair X - (X1, X2) such that X, " Li --+ Lj are maps and X,(x) ~ x ~ X,(x) for each element x c Li, then X, are isomorphisms. Proof. First, let us prove that X, (X, (x)) - x for each x c Li. Clearly,
x, (x, (~)) v x,, (x) ~ x, (x,, (*)) and, therefore,
so that x _< X, (X, (x)) and X, (X, (x)) < x i
since 4
i
Li --~" Hence X, (X, (x)) - x, X, are bijections and X,
-
X~ -1.
i
Furthermore, if (x, y) E Li x Li is any pair, then X,,(x) 4 x 4
X,,(x) and X,(Y) 4 Y 4 X~(Y)
imply
x~ (x) Aj x, (y) ~ x A~ y ~ x, (x) Aj x, (v). On the other hand since
we obtain that and so because ~ In, --<. J Finally, the case of Vi can be proved similarly and thus X~ are isomorphisms. F1 As in the first case the pair X - (X1, X2), is called the G.identity operator. C o r o l l a r y 6.1.38. Let (L - L1U L2, ~ , X) be a goset, where (L~, A~, V~, 0, e) be lattices, L1 N L2 - {O,e} and X - (X1, X2) be a G.identity operator. Define binary operations
i~; inf(~, y), i(, ~up(~, y ) ( L ~
• L~)u (L, • L , ) - ~ L~
as follows"
i~-i.f(~, y) - x A~ v, i~-sup(~, v) - x v~ y if (x, y) c L~ • L~
~
~
~
~
9
~,...~. "-"
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240
vi. Generalized Boolean Algebra and Related Problems
Conversely, if Z: is a G.lattice, then (Ls, Ai, Vs, O,e) are lattices in the usual sense, satisfying the conditions which correspond to a) and b) of (4). Therefore (Li, Ai, Vi) are Boolean algebras ([26, [29]) and thus, by (c) of Corollary 6.1.38, s is a GBA. D The pair I = (11, 12) is called the G.rejection operator. Our next result is concerned with a V-formation and the corresponding strong a m a l g a m a t i o n in the sense of [127]. Let s = {L1, AI, V1, 4 , L 2 , A2, V2} be a G.lattice. We define the binary relation (c) on L1 U L2 as follows: if x c Li, y c L 1 U L2, then x(c)y ~
y = x or y = x Vj (~.
Let us prove t h a t (c) is the congruence o n L1 U L2. In the first place, note t h a t (c) is the equivalence relation o n L 1 U L 2. (1) Suppose that x, xl, y, yl E L i and x(c)y, Xl(C)yl. Then x = y and XI --- Yl. Hence (x Ai X l ) ( c ) ( y Ai Yl) and (x Vi X l ) ( c ) ( y Vi Yl) since X Ai Xl = y Ai Yl a n d x Vi Xl = y Vi Yl.
(2) If x, Xl E Li, y, Yl E Lj, then y = x Vj I~ -- x Aj e, yl = Xl Vj (~ = Xl Aj e and we have
y Aj yl - (x Aj ~) Aj (Xl Aj ~) - (x A~ x,) Aj ~, y vj yl - (x vr e) vj (x~ vj e) - (x v~ Xl) vr e since/2 is a G.lattice so that
(x Ai Xl)(e)(y Ai Yl) and (x Vi xl)(c)(y Vj Yl). (3) If x, y C Li, Xl, Yl C L j , then x - y, Xl - yl and it follows that
(X Aj Xl)(C)(y Aj Yl), (x Vj Xl)(c)(y Vj Yl). (4) If x, yl 6 Li, X l , y C Lj, then y - x Aj e, Yl -- Xl Ai e and x Aj Xl - (x Aj ~) A~ Xl - y Aj Xl - y Aj ((~1 A~ ~) A~ ~) -
= y A j (Yl Aj e) -- (yA i Yl) Aj e. Similarly, x Vj X 1 -- (y V i Yl) Vj (~1, that is,
(x Aj Xl)(c)(y Aj Yl) and (x Vj Xl)(c)(y Vj Yl). Thus (c) is the congruence on L1 U L2. The congruence class, containing an element x E L1 U L2, is denoted by Ix], while the set of classes [x] - by s Let us define binary operations 9, U ' ( s x s ~ s as follows: [x] N [y] -- [x As y] and [x] U [y] - [x Vi y] for each pair (x,y) E L k x L i . It is not difficult to see that ( s [(9], [e]) is a lattice in the usual sense, obtained from s by sticking each element x c Li \ {(~, e} to the element x Vj (~ and the elements (9, e to themselves. Moreover, the maps
mi " (Ls, Ai, Vi, (9, e) ~ (C/c, n, I I, [0], [el), defined as m s ( x ) -
[x], are isomorphisms.
6.1. Gosets, G e n e r a l i z e d L a t t i c e s , . . .
241
Since the lattice C(2) = {L = {(~, e}, A, V} is a sublattice of any lattice which contains (9 and e, we obtain t h a t if s = { L 1 , A 1 , V I , ( ~ , 4 ,e, L2, A2, V2} is a G.lattice, then the quintuplet (s tl,t2), where t~ : s --, L~ are algebra embeddings, is the V-formation since m l O t l = m 2 o r 2 , t h a t is, the diagram ml
L1
s is commutative. Moreover, (ml, m2, C/c) since
t
this
>C/c
t2 > L2 V-formation
is strongly a m a l g a m a t e d
m , ( L 1 ) F1 m2(L2) - m l ( t l ( L ( 2 ) ) - m2t2(L(2))
by
[127].
Our consideration in this section will not be complete unless we make at least a casual mention of the real-valued functions on G B A ' s and bitopological GBA's. For this, we need to introduce a few handful notions. Suppose s {LI,A1,V1,Q},4 ,e, L2, A2, V2} is a G.lattice. Then the elements x E L1 U L2 and y c Li are disjoint, t h a t is, x d , y if x Ai y -- ~ . Furthermore, an element x 6 L1 U L2 is disjoint from a set E C_ L 1 U L 2 , t h a t is, x d c E , if x Ai y -- 0 an for each element y C E~, where Ei - E N Li. D e f i n i t i o n 6 . 1 . 4 0 . Let A = {A1, A 1 , V I , ~ I , Q } , ~ ,e, A2, A2, V2,~2} be a G B A and E = E1 U E2 c A1 U A2. T h e n E is said to be a G.disjoint system of elements of A if xd~((Ei \ { x } ) U Ej), where x c Ei is an arbitrary element, t h a t is, if x Ai y = (3 for each y c E~ \ {x} and x Aj z = (9 for each z c Ej. The notion of a G.disjoint system of elements is used to give the following D e f i n i t i o n 6 . 1 . 4 1 . Let f i [ - {A1, A 1 , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2, q~2} be a GBA. A pair p (#1,#2), w h e r e p i " Ai ~ R are finite maps, is said to be a G.quasi measure on A, if pi(x) >_ 0 for each element x c Ai and
xEE 3
yEEi
xEEj
yCE,
for every finite G.disjoint system E = E1 U E2 C A1 U A2. It is obvious t h a t if p = (pl, p2) is a G.quasi measure, then #~(0) = 0. P r o p o s i t i o n 6 . 1 . 4 2 . If p = (#1,#2) is a G.quasi measure on a G B A A, then x 4 y implies that pk(x) <_ #~(y) for each pair (x, y) ~ Ak x A~.
Pro@ First, let x, y E A~ and x < y. T h e n the elements x and y - x = y Ai ~i(x) i
are disjoint, y - x Vi ( y - x) and hence i
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6.2. Generalized Ideals and their Variety . . . .
243
i f x 4 Y, then Cli(x) 4 Clj(y) and Inti(x) 4 Inty(y) for each pair (x, y) c Ai x Aj.
(9)
x = Cl~(x) ~
~i(x) = Inti(~i(x)) z--> ~i(x) = I n t j ( p i ( x ) )
and x = Inti(x) ,z--->, ~i(x) = Cli(~i(x)) ,z----5,~i(x) = Clj(~i(x)) for each x c A i. The proof, based on elementary calculations, is omitted. A more detailed development of the questions connected to bitopological GBA's and quasi measures on GBA's, in our view, is of independent interest also.
6.2. Generalized Ideals and their Variety. Stone Family of P r i m e Generalized Ideals All our further constructions are essentially connected with the notion, introduced by
Definition 6.2.1. A G.ideal (briefly, GI) of a G.lattice/2 = {L1, A1, Vl, 1~, 4 , e, L2, A2, V2} is a pair I = (I1,/2), where Ii C_ Li and which satisfies the following conditions: (1) I f x c I l U / 2 andycIi, thenxV, ycIi. (2) If x E I1 U/2, y E L1 U L 2 and y 4 x, then y c I1 U 12. E x a m p l e 6.2.2. Let A = {A1,A1,V1,q21,(~,~,e, A2, A2, V2,~2} be a GBA and p = (p l, p2) be a G.quasi measure on A. Then it is not ditticult to see that I = ( I I , h ) , where I~ = {x c A , : p~(x) = 0}, is a GI. Moreover, note that the pairs ({a,d}, {c, d}) and ({m, d}, {n, d}) in Diagram 3 are GI's. Since for every GBA A = {A1, A1, V1, qP1, (~, 4 ,e, A2, A2, V2, P2}, the system { A 1 , A 1 , V I , ( ~ , 4 ,e, A2, A2, V2} is a G.lattice, in the sequel we shall consider, in general, GBA's.
P r o p o s i t i o n 6.2.3. Let A = {A1, A1,V1,~91,O,~ ,e, A2, A2, V2,~2} be a GBA. Then a pair I = (/1,/2), where I~ c_ A~, is a GI if and only if {I~, A1, V1, 4 , /2, A2, V2} is a G.sublattice of the G.lattice {A1, A1, V1, 0 , 4 , e, A2, A2, V2} and x 6 I1 U 12, y 6 A~ imply x A~ y E Ii. Proof. First, let I=(I1, h ) be a GI and let us prove that {/1, A 1 , V I , ~ , / 2 , A2, V2} is a G.sublattice of the G.lattice {A1,A1, V1,0, 4 ,e, A2, A2, V2}. Indeed, if x c I1 U/2 and y c Ii, then by (1) of Definition 6.2.1, z Vi y c Ii. It is evident that z Ai y 4 z and by (2) of Definition 6.2.1, x Ai y E I1 U/2 ~ x Ai y 6 I/. Thus {/1, A1, V1, 4 ,I2, A2, V2} is a G.sublattice. Now, if x c I1 U 12 and y c A~, then x Ai y 4 z and once more applying (2) of Definition 6.2.1 gives that z Ai y C I 1 U / 2 ,z-----N,x Ai y 6 I~.
244
VI. Generalized Boolean Algebra and Related Problems
Conversely, let {I1, A1, V1, ~ , I2, A2, V2} be a G.sublattice of the G.lattice { A 1 , A 1 , V 1 , O , 4 ,e, A2, A2, V2} and x E I1 U / 2 , y C Ai imply x Ai y C Ii. Let us prove t h a t the conditions (1) and (2) of Definition 6.2.1 are satisfied. Indeed, if x c I1 U / 2 and y E Ii, t h e n x Vi y c Ii since {I1,A1, V1, 4 ,I2, A2, V2} is a G.sublattice, t h a t is, (1) of Definition 6.2.1 is satisfied. Finally, if x c I1 U / 2 , y E A1 U A2 and y ~ x, t h e n if, for example, we consider the case y c A j , we obtain t h a t y - x A j y and, therefore, y c lj c I1U/2, t h a t is, (2) of Definition 6.2.1 is also satisfied. D It is obvious t h a t if I - (I1, I2) is a GI, then Ii are ideals in the usual sense and the pair I - (A1, A2) is a GI. It is likewise obvious t h a t for a GI I - (I1, I2), we have x E I1 U 12, y C Ii ~ x V i y C Ii. M o r e o v e r , I1 7s A1 ~ 12 ~= A2 and, therefore, a GI I - (I1,I2) is said to be proper if I~ ~ A~. Thus, by (2) of Definition 6.2.1, I - (I1, I2) is proper
e---->, e g I1 <-->, e c / 2 .
Hence, using (1) of the latter definition, the a r g u m e n t s between T h e o r e m 6.1.31 and Corollary 6.1.32, and (7) of Corollary 6.1.33, we obtain: I -- (I1,/2) is proper -'
<---5, P v ( ~ l ) N (11 x / 2 ) - 2~ ~' :,
,'- Pv(~2) K~(I2 x I1) - 2~ ~ -'
P v ( ~ l ) K I (I1 • I1) - 2~ ,:
:,
:- Pv(~b2)A (I2 x I2) - 2~.
Note further t h a t I ~ - ({(9}, {@}) is called the zero GI, and for any proper GI (I1, I2), we have 11AI2 - {(9}. It is not difficult to verify how I1 a n d / 2 are i n t e r c o n n e c t e d in any proper GI. Following (6) of T h e o r e m 6.1.16, for each x c Ai, we have x 4 x Vj 1~ and x Vj 1~ ~ X. Now by virtue of (2) of Definition 6.2.1, we obtain x c I~ ,<--->, x Vj (~ E Ij. Therefore any proper GI can be w r i t t e n as I - (I1 - { x } , I 2 - { x V2 (9}), where Ii C Ai so t h a t 1 I I 1 - /21. Moreover, if A1 contains an a t o m a (<---5, A2 contains the a t o m a V2 (~), then it is clear t h a t I - ({a, @}, {a V2 (9, @}) is s t r u c t u r a l l y the simplest type of a GI after the zero GI I ~ Let I - ( I I , h ) and I ' - ( I ~ , I ~ ) be any two GI's o f a G B A A. Then, by virtue of the above reasoning, I
-
I1
--
I~ <-->, I2 -- 1s and I1 C I~ <-->, h C I;.
In the former case we write I - I', while in the l a t t e r - I < I'. Hence I < I ' ,' :, (I < I ' or I - I'). Clearly, the binary relation < is a partial order on the set Z{/t - (I~,/~) " t e T} of all GI's of the G B A A. T h e zero of this poser coincides with the zero GI I ~ and the unity - with the unit GI I e - ( A 1 , A 2 ) . As usual, a subfamily {/t - (I~, I~)" t c To c T} of 2- is a chain i f / t <_/t, o r / t , <_/t for each pair t, t ~ c To. Proposition any intersection
6.2.4. For a GBA A- {A1,A1,VI,~I,0,~ ,e, A2, A2, V2,~2} o f G I ' s a n d the u n i o n o f a n y c h a i n i f G I ' s is a GI.
P r o o f . First, let { I t - ( I f , I t ~ ) " t c To c T} be any family of GI's of a G B A A. We are to prove t h a t I - (I1,/2) - ( ~ I~, ~ I t) is also a GI. Assume t h a t tETo
tETo
6.2. Generalized Ideals and their Variety . . . .
245
x 6 I1 U /2 and y 9Ii. If x 9Ii, then x ~ I t for each t ~ To and, by (1) of Definition 6.2.1, x V i y 9 I tfor e a c h t 9To so t h a t x V i y 9Ii. I f x 9Ij, then x 9 I tfor each t ~ To and, therefore, x V~ y 9 I tfor each t 9To, t h a t is, x V~ y ~ I~. Before we proceed to proving (2) of Definition 6.2.1 note t h a t
( N
( N z:) - N
t E To
t E To
t E To
t E To
t E To
o
Indeed, it is clear that
t E T~~
Hence it can be assumed t h a t x c {@, e} and x c
n (I~ u I~). In that case x c I~ t6To
or 2; C I~ for each t E T O since A1 n A2 -
{E:),e} and, t h e r e f o r e , 2; c
n
I~
tGTo
or x C n
I2t. Hence x c ( n
t 6 To
I1t) u ( n
t 6 To
I2t) 9Now assume t h a t x c I1 U I2,
t 6 To
y 6 A1 U A2 and y ~ x. W i t h o u t loss of generality let us consider the case where X 9I1 -- n I~. Then y ~ x and x 9 I~for each t c To imply x 9 I~ U I2t for each t6Tc) t 9To and y ~ x. By (2) of Definition 6.2.1, y 9 I~ UI2t for each t 9To so that y C n (1~ u/2t) -- ( n 1~) u ( n I ~ ) - I l U I 2 . t E Tu
t E Tu
t E Tu
Next, let us assume t h a t {It - (I~,It2) " t 9To C T} is a chain of GI's a n d I -- (11,22) -- ( U I~, U I~). T h e n x c I x u / 2 a n d y 9 I i i m p l y x c
( U zf)u( U zl) t E To
If x
t E To
9
t E To
t E To
.ayc U If. t 6 To
U It, then there exists an index tl
9To such t h a t x c I tx. Similarly,
t6To
there exists an index t2 c To such t h a t y 9 t2. I It is clear t h a t I~ 1 c I~ 2 or I t~ C_ I tl . W i t h o u t loss of generality let I tl C I: 2 . Then x 9 I~ ~and, therefore, x V~ y 9I~ 2 C Ii. Now assume t h a t x 9 U I~. Then there exists t l e To such t h a t y 9 I~ ~ tE Tu
sinceyE
U It and there existst2 t E To
9
such t h a t x
9
~ sincex
9 U It. It is t E To
obvious t h a t / t ~ <_/t2 o r / t 2 _
246
VI. Generalized Boolean A l g e b r a and R e l a t e d P r o b l e m s
of a GBA A and
z'-
{I~, - (I~', I~tt ).
t'
c To c T, (B,, B~) _< I~, for ~,~h t' ~ To }
then 27' r 2~ as I c - (A1, A2) E Z'. The smallest (with respect to <_) GI, containing (B1, B2), is said to be generated by the pair (B1,B2). Clearly, the GI, generated by the pair (~, ~), is the zero GI. Also, note that {27, A, V} is a lattice in the usual sense with I ~ and U as the zero and the unit element, respectively, where for I ' - (I[, I~), I " - (I~', I~') E Z, we have
I' A •
n I~', I~ n I~')
and I ' V I " is the GI, generated by the pair (I{ O I{', I~ O I~'). Before proceeding to our next important theorem, let us recall that by (6) of Theorem 6.1.16 if x, y E A~, z E Aj, then X ~ y, i
X ~ Z ,,~,---)', X \ / j (9 ~ y,
X\/j(9
~ Z. j
Hence, taking into account the fact t h a t for each GI I - (I1,I2), we have Ij X, (Ii), the following equivalences are obvious: (For each element x C I1, there exist finite sequences of elements al, a2,. 99an c A1, b l , b 2 , . . . , bm E A2 such t h a t x <_ al Vl a2 Vl "--Vl an and x 4 bl V2 b2 V2 1 ..9 V2 b,~.) .z---> (For each element x E I1, there exist finite sequences of elements al,a2,...,an c A1, b l , b 2 , . . . , b m C A2 such that x V 2 ( 3 <_ bl V 2 b 2 V 2 " " V 2 b m and 2 x V2 O 4 al V1 a2 Vl .-. V1 an.) ~ (For each element x c I1, there exist finite sequences of elements bl, b2, . . . , bm E A2, al, a2, . . . , an C A1 such t h a t x _< (bl Vl 1
(9)vI(b2VI(9)VI" "VI(DrnVI(9) a n d x
4 (al V2(9)V2(a2V2(9)V2. . .V2(anV2(9).) ." "," (For each element x E I1, there exist finite sequences of elements bl, b 2 , . . . , b~ c A2, al, a 2 , . . . , an e A , such that x V~ (3 < (al V2 (9) V2 (a2 V2 (9) V 2 " " V2 (an V9 (9) 2
and x V~ (9 4 (bl V~ (9) Vl (be V~ (9) V l - . . Vl (b,~ Vl (9).) Note t h a t by virtue of GL3 we can easily ascertain by induction that the equalities (a 1 V 2 e ) V 2 (a 2 V 2 1~) V 2 " "
V2 (a n V2 l~) -- (a 1 V 1 a 2 V l . . .
V 1 an) V 2 (~)
and
(hi Vl e) v, (b~ v, e) v , . . . v, (bm V, e) - (hi v~ b~ v~... v~ bin) v, e are valid for finite sequences of elements al, a 2 , . . . , an 6 A1, bl, b 2 , . . . , bm 6 42. T h e o r e m 6.2.5. L e t A {A1, A 1 , V 1 , ~ 1 , 0 , 4 , e , A2, A2, V2,~2} be a G B A and (B1,B2) be a pair such that Bi C A i . T h e n a pair I - (I1,I2) is a GI, w h e n e v e r it satisfies one of the following equivalent conditions: (a) I1 - {x E AI" there exist finite sequences of elements al, a 2 . . . , an C B1 and b l , b 2 , . . . ,bin E B2 such that x < al V1 a2 V 1 . . . V1 an, x ~ bl V2 1
b2 V 2 " " V2 bin}.
6.2. G e n e r a l i z e d Ideals and their Variety . . . .
247
(b) 12 - {y c A2" there exist finite sequences of elements c l , c 2 , . . . , c k and d l , d 2 , . . . ,dz E B2 such that y < dl V2d2 V2 . . . V2dl, y ~ Cl V1 2 -9. V 1 Ck}. Moreover, this CI I - (I1,/2) is generated by the pair (B1, B2) if and only each element a ~ Bi, there exists a finite sequence of elements b,, b 2 , . . . , bt such that a 4 bl V j be V j . . . V j bt.
c B1 c2 V1 if f o r c Bj
P r o @ First, we shall see t h a t I - (I1, h ) is a GI, t h a t is, (1) and (2) of Definition 6.2.1 are fulfilled. Indeed" (1) Clearly, we have the following four variants: y E I1, x c I1 or x E /2; y C 12, x C I1 o r x C 12. Since all these variants are proved by a similar scheme, we shall consider only the case x c I~, y ~ / 2 . By (a) and (b), there are finite sequences of elements a l , a 2 , . . . , a n ; C l , C 2 , . . . , c a ~ B1, b l , b 2 , . . . , b ~ ; d l , d 2 , . . . , d z ~ B2 such t h a t
x <_ al V1 a2 V 1 - . . V1 an,
x 4 bl V2 b2 V 2 " " V2 bm
y_
y% ClVlC2Vl...VlCk.
,
and 2
Therefore 2
and Let el al,e2 a2,...,en b , , . . . , g,~ - b,~,gm+l - d l , . . . , elements e l , e 2 , . . . , e n - b k C B 1 virtue of which x V2 y c I2. (2) Let x c I1 U I2, y c A1 -
-
-
-
x c I1,
-
y c A,
-
an, en+l Cl,...,en+k ck and gl bl,g2 g,~+z - dz. It is clear t h a t the finite sequences of and g l , g 2 , . . . , g m + l C B2 are those sequences by -
-
-
-
-
-
-
-
U A2 and y d x. Here we also have four variants:
or y c A2;
x c I2,
y c A1 or y c A2.
We will prove only the case where x c I2, y E A1 and y d x; the others can be proved similarly. By (a), there exist finite sequences of elements a l , a 2 , . . . , a , ~ c B1, b l , b 2 , . . . , b m E B2 such t h a t x < bl V2 b2 V2 . . . V2 bm and x d al V1 a2 V1 ..- V1 an. 2
It is obvious t h a t y _~ al V, a2 V1 . . . VlaN and y ~ b, V2 b2 V2 .-. V2 b~ 1
so t h a t y E I1. Thus I = (I1,/2) is a GI. Now let us prove the second part of the theorem. If the GI I = (I1,/2), constructed above, is generated by the pair (B1, B2), then Bi C_ Ii and, therefore, the "only if" part is obvious. Conversely, let there exist for each element a c B~, a finite sequence of elements bl, b 2 , . . . , bt c By such t h a t a ~ bl Vj b2 Vj . . . Vj bt. Then it is clear t h a t a E Ii and, therefore, Bi C_ Ii. Hence, it suffices to prove
248
VI. Generalized Boolean Algebra and Related Problems
that I - ( I i , h ) is a smallest GI such that ( B I , B 2 ) <_ I ,e---->, Bi c_ Ii. Indeed, if I ' - (I~, I~) is any GI for which (B1,B2) <_ I' and x E h is any element, then by assumption, there exists a finite sequence of elements a l, a 2 , . . . , an C Bi such that x <_ al Vi a2 Vi . . . Vi an. Clearly, a l , a 2 , . . . , a n C B i C_ I~ implies x E I~ so i that Ii c_ I ' because x E Ii is an arbitrary element and, therefore, I _< I'. D This theorem gives rise to the following interesting C o r o l l a r y 6.2.6. Let A - {A1, A1, V I , ~ I , 0 , 4 , e , A2, A2, V2, qD2} be a GBA and let I - (11, I2) be the GI generated by a pair (B1,B2), where Bi c Ai. Then the following conditions are equivalent: (1) I - (I1, I2) is nonproper. (2) There is an element a E B1 for which there ezist finite sequences of elements a l , a 2 , . . . , a n E B1, b l , b 2 , . . . , b , ~ E B2 such t h a t p l ( a ) ~ al V1 a2 Vl -.- Vl an and ~ l ( a ) _< bl V2 b2 V2 ... V2 b,~. 2
(3) There is an element b E B2 for which there exist finite sequences of elements C l , C 2 , . . . , c k c B1, d l , d 2 , . . . , d l c B2 such that ~2(b) ~ dl V2 d2 V2 . . . V2 dz and ~2(b) <_ cl Vl c2 V 1 . . . Vl ck. 1
(4) There is an element c E B1 for which there exist finite sequences of elements e l , e 2 , . . . , e t E B1, g l , g 2 , . . . , g r C B2 such that ~bl(C ) 4 gl V2 g2 V2"'" V2 gr a n d ~bl(C ) ~ el V1 e2 V I " " V1 et. 1
(5) There is an element d E B2 for which there exist finite sequences of elements P l , P 2 , . . . , P u c B1, q l , q 2 , . . . , q v E B2 such that r 4 Pl ~/1 P2 V1 . " VlPu and r < ql V2 q2 V2 ... V9 qv. 2
Proof. Applying Theorem 6.2.5, in four cases of the existence of finite sequences of the elements of B1 and B2, we respectively obtain ~ l ( a ) c I2, ~2(b) c 11, r E I l a n d ~ b 2 ( d ) c / 2 . T h e r e f o r e a , c c B1 C_ I1, b, d E B2 C_ /2 and (1) of Definition 6.2.1 imply that a V 2 q ~ l ( a ) - - e e I 2 , bV1 ~ 2 ( b ) - e e I 1 ,
cV1 ~ b l ( c ) - e e I 1 , and d V 2 ~ b 2 ( d ) - e e I 9
so that I - I e - (A1, A2) in each one of the four cases. Conversely, let I - I ~ - (A1,A2). Then e c I1 - A1, e E / 2 - A2. Hence, by virtue of Theorem 6.2.5, there exist finite sequences of elements al, a 2 , . . . , an E B1, bl, b2,... ,bin E B2 such that e - al V1 a2 Vl . . . V1 an -- bl V2 b2 V2 "" V2 bin. Clearly, for each element a E B1 and each element b E B2, we have ~ l ( a ) 4 al V1 a2 V 1 . . . V1 an, ~l(a) < bl V9 b2 V2-.. V2 bm; 2
~2(b) 4 bl V2 b2 V2.-. V2 bin, g)2(b) <_ al V2 a2 V 2 . . . V2 an; 1
~bl(a) 4 b~ V2 52 V 2 . ' - V 2 bin, ~bl(a) _< al V1 a2 V I ' " V1 an; 1
@2(b) 4 al V1 a2 V I " " V1 an, ~2(b) _< bl V2 b2 V2"'" V2 bin. 2
~]
Thus for the GI I - (I~, I2), generated by the pair (B1, B~), to be nonproper it is sufficient that there exists at least one element of the set B1, satisfying the
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252
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
and
{xv =
{y ~ A j "
e.
xe
there exists an element y~ ~ I~ such that y ~ z V~ y~},
is a GI. Moreover, this GI is generated by the GI I' - (I~,I~) and the element z 6 Ai \ I~ (so that I - (I1,I2) is the smallest GI for which I' < I and z E Ii) if and only if for each element y' c I}, there exists an element x E I~ such that y'4x. Pro@ First we are to prove t h a t I - (I1,/2) is a GI. Since in our concrete case z E A~ \ I ' in proving (1) of Definition 6.2.1 we shall consider b o t h y E I~ and ycIj. If y E Ii, z c I1 U/2, t h a t is, if x E I i or x E Ij, then in b o t h cases, we obtain __ z Vi y < z Vi (x' Vi y'), where x' Vi y' E I~ since x', yl E I il and I iI is an ideal in i the usual sense. Therefore, by condition, x V i y E I i . On the other hand, if y E I o, x E I~ or z c Ij, then in a similar manner, we obtain x Vj y 4 z V~ (x' V~ y'), where z' Vi y' E I~, and, therefore, x Vj y c Ij. Let us prove (2) of Definition 6.2.1. For this we assume t h a t x E I1 U / 2 , y 6 A1 U A2 and y 4 x. We shall consider only the cases x E Ii, y c Aj and z E Ij, y E Ai since the other cases are proved quite similarly. For x c Ii and y 6 Aj, we obtain y 4 z Vi z', while for x E Ij, y E Ai, we obtain y _< z Vi z', i
where x' c I[. Therefore y c Ii. Now, let us prove the second part. If a GI I - (I1,/2) is generated by a GI I ' - (I~ , I~) and an element z E Ai \ I '~, then I ' < I and, by the first part, for each element y' c I5 there exists an element y" E I~ such t h a t y' 4 z Vi y". Hence the conditions I~ c Ii and z E Ii imply z V~ y" c Ii, t h a t is, z - z Vi y" is the required element. Conversely, let us assume t h a t I - (I1,I2) is the GI, constructed using the GI I ' - (I~, I~) and the element z E Ai \ I~. It is clear t h a t I~ c Ii and z c Ii. If y' c I}, then the existence of an element x c Ii with the condition y' 4 z implies y' E Ij and, therefore, I~ c Ij since y' is an a r b i t r a r y element of I5. Thus I ' < I and it remains only to prove t h a t if I " - (I{', I~') is any GI such t h a t I ' < I " , and z c I~', then I _< I " . Indeed, for an a r b i t r a r y element x E Ii there is an element x~EI~suchthatx<_zVix ~ ButI ~
x E I~~. Similarly, if y c Ij, then there is an element y' c I~ such t h a t y 4 z Vi y'. Since z Vi y~ C I "i , we obtain y E I~t and so I _< I". D Note that, by virtue the first part of T h e o r e m 6.2.7, for the GI I - (I1,/2), we can write the structure of Ij in more precise terms as follows:
{xv -
{y E Aj - {y c A j "
e-
xc I,} -
9there exists an element y' E I~ such t h a t y 4 z Vi y'} there exists an element t' E I} such t h a t y <_ z V j t'}, J
where obviously t ~ - y' Vj 0 .
6.2. Generalized Ideals and their Variety . . . .
253
C o r o l l a r y 6.2.8. Let A = { A 1 , A I , V I , ~ I , ( ~ , ~ ,e, A2, A2, V2,~2} be a GBA, I = (I1, I2) be the GI, generated by a GI I ' = (I~,I~) and an element z ~ Ai \ I~. Then the following equivalences hold:
I = (h, h)
e Ij
9
Proof. Since, by (1) of Corollary 6.1.33, ~ = ~j o X,, (2) of Corollary 6.1.32 and (2) of Theorem 6.1.16 imply X, o X:, o ~ i = X, o ~Pi ~
9~i = X, o ~'i,
and, hence, if x c Ai is any element and I = (I1,/2) is any GI, then c
x,
c 6.
Thus it remains to prove the first of our equivalences. If ~ ( z ) c I}, then I' < I and z r imply that I = I e = (A1, A2). Conversely, let I = I e = (A1,A2). Then there exists an element x I E I~ such that e = z Vi x'. Therefore ~i(e) = ~i(z) Aj ~i(x') = O and by (5) of Corollary 6.1.32, ~ i ( z ) 4 x'. Thus ~ i ( z ) c I~ since x' c I~. rq Now, we have come to the most important variety of GI's which will be essentially used in our further reasoning. A GI I = (I1,I2) of a GBA is said to be maximal if it is proper and has no property to be contained in a proper GI of A so t h a t there does not exist a proper GI I ' - ( I ~ , I~) such t h a t I < I'. D e f i n i t i o n 6.2.9. A GI I = (/1, I2) of a GBA .4 = {A1, A1, V1, ~ 1 , (~, 4 , e, A2, A2, V2, ~2} is said to be prime if it is proper and the following condition is satisfied: if x E A1 u A2, y 6 Ai and x Ai y C Ii, then either x r I1 U/2 or y r Ii (or both x c 11 U 12 and y c / ~ ) . Therefore, if I = (I1,I2) is a prime GI and x c Ai is any element, then x e Ii or pi(x) e Ij(..v---> ~i(x) = X, (pi(x)) e Ii), but not both, because the GI I = (I1,/2) is proper. On the other hand, if there exists an element x c Ai such that x c I~ and V)i(x) c Ij, then x Aj ~ ( x ) = (9 C Ij and to obtain a contradiction, it suffices to apply the condition of Definition 6.2.9. T h e o r e m 6.2.10. Let I = ([1,12) be a GI of a GBA A = {A1, A1, Vl, (/91, (~), ~---~, e, A2, A2, V2, ~2}. Then the conditions below are equivalent: (1) I = (I1, I2) is a prime GI. (2) For every element x c Ai either x E Ii or pi(x) E I j ( < - - > r x. c both
=
Pro@ The implication (1) ----5, (2) is shown in the discussion preceding Theorem 6.2.10. (2) ~ (1). By (2) it is obvious that the GI I = (I1,/2) is proper. Now assume t h a t x, y c A~, x A~ y C/~ and x ~ 5 , YE I~. Then by (2), ~i(x), ~ ( y ) c Ij and by ( 1 ) o f Definition 6.2.1 and ( 2 ) o f Theorem 6.1.31, we have F~(x)Vj F ~ ( y ) = p~(x A~ y) ~ Ij. Therefore by (2), x A~ y r I~, which is impossible. The case x ~ Aj,
254
VI. Generalized Boolean Algebra and Related Problems
y 9A~ and x A~ y 9I~ is proved in a similar manner, taking into account (1) of Definition 6.2.1 and (3) of Theorem 6.1.31. (1) - - > (3). Let the prime GI I = (I~,I2) be not maximal. Then there is a proper GI I ' - (I~,I~) such t h a t I < I'. I f x 9I ~ \ I i is any element, then (1) <---> (2) implies t h a t ~ ( x ) 9 Ij so t h a t ~ ( x ) 9 I5and, therefore, P v ( ~ ) N (I~ • I5) ~ ;~, which contradicts the condition t h a t I' - ( I { , I ~ ) is a proper GI. Hence I = (I1,I2) is maximal. (3) ---->. (1). Let x 9A1UA~, y 9A~ and x A ~ y 9I~. W i t h o u t loss of generality assume t h a t x 9A~ \ I~ and consider the GI I ' - (I~, I~), generated by I - (I1, I2) and x-gI~. Clearly, I ' = I ~ = ( A I , A 2 ) since I = (I1,I2) is maximal. Therefore for e 9Ai, there exists an element z 9I~ such t h a t e = x V~ z. Hence
sincez
9
andyAiz
If we now assume t h a t x 9A j \ Ij and I ' - (I[, I~) is generated by I - (I1, I2) and x - g I j , then I' - I ~ - ( A 1 , A 2 ) and thus for e 9A j , there exists an element z 9Ij such t h a t e - x Vj Z. Hence, by GDL2, we have
y - (x v j z)
y - (x
y)
(z
y) e
sincez 9 a n d z A i y 4 z. Therefore x g [1 U/2 and x Ai y 9Ii imply y 9Ii. Further, assume t h a t x A~ y 9I~ and y-gI~. First, let x 9A~. If I ' - (I~,I~) is generated by I = (I1,I2) and y g / ~ , then I ' = I e = (A1,A2) and, by analogy with the above, for e 9A~ there exists an element z 9Ii such t h a t e = y Vi z. Therefore Similarly, if x 9A j , then I ' = I e = (A1, A2), where I ' is generated by I = (I1, I2) and y g Ii. Hence, by the remark between Theorem 6.2.7 and Corollary 6.2.8, there is an element z 9Ii such t h a t e = y Vj (Z Vj {~). But by the G.commutativity laws, x Ai y 9Ii .e----->.(x Ai y) Vj ~) = (y Aj x) 9Ij and, therefore, x sincez
Aj
9
9 - (v
z) Aj x - (y Aj x) v j (z Aj x)
zimplyzAjx
9Ij
9
P r o p o s i t i o n 6.2.11. For a GBA A1 - {A1, A1, V1, ~1, ~, ~, e, A2, A2, V2, p2} any GI I - (I1, I2) is p r i m e if and only if the goset (A1 U A2, ~ ) is a G.chain. Proof. Let any GI of A be prime and, for example, there are elements m E A1, y E A2 such t h a t x ~ y is false and y ~ , is false. Then by Theorem 6.2.12 and Corollary 6.2.13 below there are prime GI's I x - ( I ~ , I ~ ) and Iy - ( I ~ , I ~ ) such t h a t x g I ~ , y c I~ a n d y g I ~ , x E I~. S i n c e x A 2 y ~ x, x A 2 y < y, by (2) of 2
Definition 6.2.1, x A 2 y I x Cq I v. Then x A2 y Let (A1 tO A2, 4 ) xEAltOA2, yEA~.
e I~ and x A 2 y e I y. Let I - ( I i , I u ) - ( I ~ A I ~ , I ~ A I ~ ) -C I2, but x g I1 and y g / 2 , which is impossible. be a G.chain, I (I1,I2) be a GI and x Ai y C Ii, where Thenx4 yory4 x. If y 4 x, t h e n x A ~ y - y E I ~ , a n d i f
6.2. Generalized Ideals and their Variety . . . .
255
x 4 y where, for example, x 9Aj, then x Ai y 9Ii implies y Aj x -- (x Ai y) Aj e -x 9Ij so that in both cases I - (11, I2) is prime. V] Our next theorem underlies many further constructions. T h e o r e m 6.2.12. For a GBA A - {A1, A1, Vl, ~1, (~, ~ , e, A2, A2, V2, ~2} and each element x 9Aj \ {(9}, there exists a prime GI I* - (I~,I~) such that
Proof. Let A4j - {I - (11,/2)} be a family of all GI's such that x - ~ I j and ~j(x) 9Ii(~=> ~j(x) 9Ij), where x 9Aj \ {(9} is an arbitrarily fixed element. Clearly, A4j ~ ~ since, in particular, the right principal GI (~I(X))I(R)--(II--{ycA
1 " y ~ ~l(X)}, I2-{zcA
2
9 z ~(~l(X)}) 2
9
and the left principal GI
(992(x))i(L)-(Ii-{yCAl
" y ~
1
992(x)}, I 2 - { z
9
" z 4 ~2(x)}) 9
for x 6 A1 \ {(~} and x 6 A2 \ {(~}, respectively. Following Proposition 6.2.4, every chain in 3dj has an upper bound in A4j since the family Adj is partially ordered by the relation <. Therefore by Zorn's lemma, M j has a maximal element I* - (I~, I~). Let us prove that I* - (Ii~, I:~) is the required prime GI. Clearly, x - c I ] and ~ j ( X ) 9 I[, that is, ~)j(X) 9 I~. By virtue of Theorem 6.2.10, it suffices to prove that I* - (I~, I~) is maximal. Let I' - (I~,I~) be a GI such that I* < I'. It is obvious that ~j(x) 9 I~(~' ",, @(x) 9Ij). Also, x 9Ij since if x-E I j, then I' c A4j, which is impossible by the definition of the family 34j and the definition of I* as its maximal element. Therefore P v ( ~ j ) n I} x I~ r ~ so that I ' - I e - ( A 1 , A 2 ) and thus I* - (I~,I~) is a maximal, that is, a prime GI. D C o r o l l a r y 6.2.13. For a GBA A - {A1, A1, V1, 7~1, O, ~ , e, A2, A2, V2, P2} the conditions below are equivalent:
(1) (2)
For each element x c A~ \ {@}, there exists a prime GI such that x-c I~. For each pair (x, y) E (A~ x Aj) \ PA(P~), there exists a prime GI I = (I1,12) such that x -c I~ and y -~ Ij. (3) For each pair (x, y) c Ai x Aj, where x ~ y is false, there exists a prime GI I - (/1, I2) such that x-~ I~ and y E Ij. (4) For each pair (x, y) c (A~ x A~) \ PA(~P~), there exists a prime GI I = (I1, I2) such that x -c Ii and y -c Ii. (s) For each pair (x, y) c A, x A~, where x < y is false, there exists a prime GI such that x-E Ii and y c I~.
i
Pro@ (1) ~ (2). Let (x, y) c (Ai x Aj) \ PA(~i) be any pair. Then by (6) of Corollary 6.1.32, (y,x) c (Aj x Ai) \ PA(~j) and, therefore, y Ai x C Ai \ {@}. By (1), there exists a prime GI I - (I1,I2) such that y Ai x-~Ii and thus, by (2) of Definition 6.2.1, x c Ii and y -c Ij.
256
VI. Generalized Boolean Algebra and Related Problems
(2) ~ (3). Let (x, y) E A~ x Aj be a pair such that x 4 Y is false. Then by (5) of Corollary 6.1.32,
x A~ 9~j(Y) r 0 and (x Ai 9~j(Y), e)
9(Ai x Aj) \ PA(9~i).
Therefore, by (2), there exists a prime GI I - (I1,/2) such that x Ai ~j(y)-EIi so that ~i(x Ai ~j(y)) -- p~(x)Vj y 9Ij since I - ( I i , h ) is a prime GI. Hence 9~(x), y 9Ij and x gI~. (3) --->. (4). Let (x,y) 9(Ai x A i ) \ P~(gi) be any pair. Then, by (7) of Corollary 6.1.33, y ~ 9~(x) is false and, therefore, by (3) above, there exists a prime GI I - (I1,/2) such that y ~ / ~ and 9~(x) 9Ij so that x c/~. (4) ~ (5). Let (x,y) 9A~ xA~ b e a p a i r such that x_< y is false. Then, i
by (12) of Corollary 6.1.33, x A~ r r ~ and by (7) of the same corollary, we have (x A~ r e) 9(A~ x A~) \ PA(~). Thus by (4), there exists a prime GI I - (I1, Is) such that x Ai @i(Y)C Ii so that @i(x A~ t~i(Y)) -- t~i(x) Vi y
9Ii
since I - (I~,/2) is a prime GI. Hence ~bi(x), y 9I~ and x~I~. (5) ~ (1). Let x 9Ai \ {O} be any element. Then x _< ~Pi(x) is false and by (5), there exists a prime GI I - (I1, Is) such that x-~I~.
i
[3
C o r o l l a r y 6.2.14. I f { I t - (I~,I~)}tcT is a family of all prime GI's of a GBA then ( ~ I~, r] i ~ ) - ( { e I , { e } ) .
.A-{A,,A1,V1,9~,,O,~,e, A2, A2, V2,9~2},
tET
tCT
Proof. The proof is an immediate consequence of Theorem 6.2.12.
[3
Thus the intersection of all prime GI's of a GBA is the zero GI. However, we shall see that every infinite GBA (i.e., A1 and, therefore, As too is infinite) has families of prime GI's the intersection of which is the zero GI and which do not contain all the prime GI's of ,4. The following definition will be of much use. D e f i n i t i o n 6.2.15. A family S - {I - (I1,I2)} of prime GI's of a GBA A - {A1, A1, V1,9~a, O, 4 , e, A2, Ae, V2, 9~2} is called a Stone family of prime GI's if ( A t 1 , AI2 ) -- ({(~}, ( ( ~ } ) -- I O. IEs
T h e o r e m 6.2.16. Let S be a family ofprime GI's of a GBA A = {A1, A1, V1, 9~1,O, ~ , e, A2, A2, V2,9~2}. Then the conditions below are equivalent:
(1) S is a Stone family. (2) For each element x E A~ \ {(9}, there exists a prime GI I such that x -~ I~. (3) For each pair (x,y) c (Ai x Aj) \ PA(9~i), there exists a (I1, I2) E S such that x -E I~ and y -EIj. (4) For each pair (x, y) c A~ x Aj, where x ~ y is false, there GI I = (I1, I2) c S such that x-~ I~ and y E Ij. (5) For each pair (x, y) E (Ai x Ai) \ PA(~i), there exists a (I1, I2) c S such that x -~ Ii and y -~ Ii.
= (I1, I2) E S
prime GI I = exists a prime prime GI I =
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258
VI. Generalized Boolean Algebra and Related Problems (7) (8) (9) (10)
/ f x = i a - s u p B , then ~ ( x ) = j a - s u p C d a B for any subset B C A1UA2. C d c B is a GI for any subset B c A1 U A2. Cdaax(L) = qPl(a)I(R) if a E A1, and CdaaI(R) = ~2(a)i(L) if a E A2. IflC = {B} is a non-empty family of subsets of A1UA2, then ~ C d a B =
Cd~
U
B61C
B61C
B.
(11) x = / c - s u p B if and only if x E Cd(;(Cdc~B ) and ~ ( x ) ~ C d a B for any
subset B c A1 U A2. Proof. (1) and (2) are immediate consequences of the respective definitions. (3) If x c u c ( B ) N A i , then b <_ x ~ ~ ( x ) A ~ b - O if b E Bi, and
i
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Cdc;(Cd~B) - {x c A1
U A2
b 4 x if
" xdGCdGB}
and, therefore, if b E Bi is any element, then x Ai b - O ~
b Ak x -- O for each element x E Cd(;B and so, we have
bdGCdaB ~
b E C d a ( C d c B ).
(5) First, let us prove that if B - B1 U B2 has an i c - s u p B , then for each element y c A1 U A2, we have:
yAiia-SUp(BlUB2)--ia-SUp((
U ( y A I a ) ) U ( U (yA2b))). aEB1
Indeed, let y c Aj be any element.
b6Bg,
< y A i i a - s u p B for each i element x c Bi and y Aj z ~ y Ai ia-sup B for each element z E Bj. Therefore
ia-stlp ( (
U (yAla))U
(
a6B1
Then y A i x
U (yA2 b ) ) ) ~ y A i i a - s l l p ( B 1 U B 2 ) . b6B2
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u
u( u
aEB1
bEB2
be any element. Then for each x E Bi, we have
x <_ x v~ ~,(y) - (y v~ ~j(y)) A~ (x v~ ~,(y)) - (y A~ x) V~ ~j(y) <_ z V~ ~,(y).
i
i
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ic-sup(B 1 U B2) _< z Vi r
i
and y Ai iG-sup(B1 U B2) <
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260
VI. Generalized Boolean Algebra and Related Problems
(11) Let x = i c - s u p B . By (5), ydcx for each y E CdoB. Therefore y A i x = (9 implies x Ak y = (9 for each y C C d c B and so x E C d o ( C d c B ). Further, by (7), pi(x) = jc;-sup B and, by (5), qpi(x) Ak y = 19 for each y E C d o ( C d o B ) . But by (4), B C C d c ( C d o B ) so that ~i(x)Akz = (9 for each z C B. Hence ~i(x) c C d c B . Conversely, if ~i(x) c CdoB, then by (3), we have x E u o ( B ). If z c u o ( B ) is any element, then applying once more (3) gives that ~k(z) E C d c B , that is, x At ~k(z) = (9 since x C C d o ( C d c B ). Therefore x 4 z and thus x = i t - s u p B since z c u c (B) is an arbitrary element. [-1 D e f i n i t i o n 6.2.20. Let A = { A 1 , A 1 , V I , q O l , ( 9 , ~ ,e, A2, A2, V2,~2} be a GBA and B = B1 O B2 c A1 U A2. Then B is said to be a G.component of A if B = C d o ( C d c B ) , that is, if C d c ( C d ~ B ) c B. Now we can formulate the following important statements. T h e o r e m 6.2.21. For a GBA .4 the conditions below are satisfied:
=
,e, A2, A2, V2, p2},
{A1,A1,VI,~I,O,~
(1) Every G. component is a GI. (2) Left and right principal GI's are G.components. (3) C d a B is a G.component for every subset B = B1 U B2 c A1 U A2.
(4) If
= {E}
a
family of a. ompo
t , th n ('1 E EC tC
al o
a G.component and thus for every subset B = B1 U B2 c A1 U A2 there exists a smallest G.component A B, containing B. (5) AB = C d a ( C d c B ) for every subset B = g 1 U B 2 C A1 u A2 and, hence, AB is a GI. Proof. Assertion (1) follows directly from ( 8 ) o f Theorem 6.2.19. (2) By (9) of Theorem 6.2.19, CdGa~(c) = pl(a)~(n), and hence Cdc;(Cdc. a l ( L ) ) = C d c ( g p l ( a ) l ( R ) )
= ~92(gPl(a))i(L) = de(L)
if a c A1. Similarly, Cdc(Cdoal(R)
) = Cdc(g)2(a))l(L) = ~l(~2(a))l(R)
= al(R)
ifaEA2. Thus we can conclude that the converse of (1) above is true for the left and right principal GI's. (3) By ( 4 ) o f Theorem 6.2.19, C d o B c C d a ( C d a ( C d o B ) ) . On the other hand since, by (4) of Theorem 6.2.19, B C C d c ( C d c B ) , it follows from (1) of the same theorem that Cd~.(Cd~(Cd~B)) c Cd~B. (4) If E0 = N E, then by (10) of Theorem 6.2.19, EC/C
Zo- [') cd (cd z)ECtC
U CdGZ
EC1C
and it remains to use (3). (5) By (4) of Theorem 6.2.19, the set C d a ( C d a B ) is a G.component, containing B, and hence AB C C d a ( C d a B ) . On the other hand, by (1) of Theorem 6.2.19,
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262
VI. Generalized Boolean Algebra and Related Problems
Note t h a t the pairs ({m, b}, {n, b}) and ({a, b), {c, b}) in Diagram 3 are GF's. In the sequel we shall consider GBA's. It is evident t h a t if F = (Fx,F2) is a GF, then Fi are filters in the usual sense and the pair F = (A1,A2) is a GF. It is likewise obvious t h a t for a GF F = (F1,F2), we have x c F1 U F2, y C Fi ~ x A i y E Fi. Furthermore, it is clear t h a t F1 r A1 ,z----5, F2 r A2, and hence a GF F = (F1, F2) is said to be proper if Fi :/: Ai. Therefore, by (2) of Definition 6.3.1, a GF F = (F1, F2) is proper ,z----5, (~ c F1 ,z----5, (9 c F2 and thus, taking into account (1) of Definition 6.3.1 and using the arguments between T h e o r e m 6.1.31 and Corollary 6.2.32, as well as (7) of Corollary 6.1.33 of the same theorem, we find t h a t F - (F1,F2) is a proper *', )" P A ( ~ 2 ) ( ]
GF <--5. P A ( ~ I ) A
(/!72 x F 1 ) - ~ ~ ",
( F 1 x/t72) -- Z ,(
PA(~)I)N (F 1 x F1)-
Z .'
,','.
" PA (~P2) C~ (F2 x F 2 ) - 2~.
Note t h a t the GF F ~ - ({e}, {e}) is called the unit GF, and for any proper G F F - (F1, F2), we have F1 • F2 - {e}. By (2) of Definition 6.3.1, it is clear t h a t every proper GF has the form F - (F1 - {z}, F2 - {x V2 O}), where x r (3, and hence Fll - IF2[. Let F - (F1, F2) and F ' - (F~, F~) be any two G F ' s of a G B A A. T h e n by virtue of the above arguments F1 - F[ ~ F2 - F~ and F1 c F~ z--->, F2 c F~. In the former case we write F - F ~ and in the latter c a s e - F < F ~. Therefore F < F ~ ,z--5, ( F - F ~ or F < F~). It is obvious t h a t the relation _< is a partial order on the set ~ {Ft - (F~,F~) 9 t cT} of all GF's of the G B A A. The zero element of this poset is the unit GF, while the unit element is the zero GF F - F O - (A1,A2). As usual, a family {Ft - (F~, F~)" t E To c T} is a chain if Ft <_ Ft, or Ft, <_ Ft for each pair of indices t, t ~ c To. A G F F - (F1, F2) of a G B A .4 is said to be maximal or a G.ultrafilter if it is proper and has no property to be contained in a proper GF of A so that, there is no proper GF F ' - (F~, F~) of .4 such t h a t F < F'. Definition 6.3.2. A G F F - (F1, F2) of a G B A .4 - {A1, A1, V1, ~1, ~ , 4 , e, A2, A2, V2, ~2} is said to be prime if it is proper and the following condition is satisfied: if x E A1 U A2, y E Ai and x Vi y E Fi, then either x E F1 t2 F2 or y c Fi (or both x c F1 t2 F~ and y c Fi).
By analogy with the reasoning after Definition 6.2.9, the notion of a prime GF immediately implies t h a t if x c A~ is any element, then x c Fi or ~ ( x ) c Fi(< :. ~2~(x) - Xj(~i(x)) E I~), but not both since the GF F - (F1, F2) is proper. An especially useful assertion is T h e o r e m 6.3.3. For a G B A A - {A1,A1, VI, qPl, ~), 4 ,e, A2, A2, V2, ~2} the following condition (the G.duality) holds: I - (I1, h ) is a GI of A if and only if F-(~2(12),~1(/1))
i8 a G F of,A.
Pro@ It suffices to prove only the implication from left to right since the proof in the opposite direction can be carried out by the same scheme, taking into account the equality ~ i ( ~ j ( I j ) ) - Ij.
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264
VI. Generalized Boolean Algebra and Related Problems
(7) Let F = (F1, F2) be the GF generated by a pair (B1, B2), where Bi C Ai. Then the following conditions are equivalent: (a) F = (F1, F2) is nonproper. (b) There is an element a c B1 for which there exist finite sequences of elements al,a2,. .. ,an E B1 and bl,b~, . . . ,bin ~ B2 such that al A1 a2 A1 "" 9 A1 a n ~ ~ l ( a ) and bl A2 bz A 2 " " A2 bm < ~ 1 ( a ) 2
(c) There is an element b c B2 for which there exist finite sequences of elements C1~ C2~ 9 9Ck9 ~E B1 and d l , d 2 , . . . ,dz E B2 such that dl A2 d2 A2 ".9 A2 dz ~ ~2(b) and C l A l C 2 A l . . . A l C k <_ ~2(b). 1
(d) There is an element c C B 1 for w h i c h there exist finite sequences of elements e l , e2 ~ 9 9e t 9 E ~ B1 and gl, g2, 9 , g~ 9 9E B2 such that gl A2 g2 A2 " ' " A2 g r ~ 1~1 (C) and e l A1 e2 A1 " ' " A1 et 1/21(C) 9 1
(e) There is an element d E B2 for which there exist finite sequences of elements P l , P 2 , . . . ,Pu E 81 and ql,q2,... ,qv E B2 such that pl A1 P2 A1 " ' " A1 P u ~ ~ 2 ( d ) and q l A2 q2 A 2 " " A2 qv ~ 1/)2(d).
(8)
If a E A1, b E A2 and
F-(F1
-{xEA
1
9a < _ x 1
81
{a}, B2 -- {b},
--
2
then
andb~ x},F2-{yEA2"
the
pair
b < y a n d a ~ y}) 2
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(FI-
{xcAI"
a<_x}, F 2 - { y c y c A 2 "
aV20
1
2
(9) If ~ ~ B1 C A1, B2 - - ~ , then the GF, generated bythe pair (Bl, Z), has the form: F = (F1 = {x E A1 : there exists a finite sequence of elements a l , a 2 , . . . , a n c B1 such that al A1 a2 A 1 . . . A1 an < x}, F2 = {y E A2 : 1
there exists a finite sequence of elements a~,a~2,... ,a fk c B1 such that a~ A1 a~ A I " " A1 a~ ~ y}) and if B 1 - - ~ , ~ ~ B 2 C A2, then the GF, generated by the pair (~,B2), has the form: F = (F1 = {x c A1 : there exists a finite sequence of elements b~l,b~,..., b~ E B2 such that b~l A2 b~ A 2 . . . A2 b~m ~ x}, F2 - {y E A2 " there exists a finite sequence of elements bl, b2,..., bz c B2 such that bl A2 b2 A2 ... A2 bz _< y}).
(10)
2
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--
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a 4 x}, F2 - {y E A2" x C A 1 } , F2 - {a V2 y"
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266
VI. Generalized Boolean Algebra and Related Problems
(16) Let F -
(F1,F2) be the GF generated by a GF F ' element z E Ai \ F~. Then F-
(F1,/72)
is nonproper
~
~i(z) c I~ ~
(F~,F~) and an
~i(z) c 1~.
(17) I f .T -
{ F - (F~,F2) 9 F is a GF of a GBA A}, then {U,A,V} is a lattice in the usual sense with F ~ and F ~ as the zero and unit elements, respectively, where f o r F ' - ( F ~ , F ~ ) and F " - ( F ~ ' , F ~ ' ) , we the pair (F~ U F[', F~ U F~').
Proof. Conditions (1)-(17) are immediate consequences of the G.duality and the corresponding statements for GI's. D
T h e o r e m 6.3.5. Let A - {A1, A1, V l , 991, 1~, ~ , e, A2, A2, V2, ~2} be a GBA, ( I i , h ) be a GI, F - (F1,F2) be a GF and I n F - 2J. Then there exists a prime GI I* - (I~, I~ ) such that I <_ I* and I* n F - ;~. I-
Proof. Let 3,t - { I ' - (I~, I ; ) " I < I ' and I ' n F - ~}. It is clear that 34 r ;~ since I E A/t, and Ad is partially ordered by _<. If C - {It - ( I ~ , I t ) " It c All} is any chain in AA, then by Proposition 6.2.4, I " - (UI~, U/t) is a GI. Therefore, by Zorn's lemma, A/l has a maximal element I* - (Ii~, I~). Clearly, I _< I* and I* N F - ;~ so that, it suffices to prove only that I* is prime. Contrary: if I* is not prime, then there exist elements a E (A1 U A2) \ (Ii~ U I~) and b E Ai \ I[ such that a Ai b E I~*. Without loss of generality let us suppose that a c A1 \ I~ and b c A2 \ I~. Since I* is a maximal GI having the property I* n F - ;~, we have a I * n F r 2~ r I*b N F and, therefore, a I[ N Fi r ;g r I[b N Fi. Let, for example, x c aIi~ n F1 and y C I~ b n/7'2. By the definition of aI* and I *b, there exist elements m E I i ~, p c Ii ~ such that x-mVla,
y-pVpb
and m V l a E F 1 ,
pV2bEF2.
Thus t - (m V~ a) A2 (p V2 b) E F2 since F is a GF. By I GDL1 and III GDL2, we have: t
--
=
((?Tt
Vl
a) A2 p)
v)
V2 ((m V1 a) A~ b) p) t,) (a -
t,).
By condition aA2b C I~. Since I* is a GI, it is clear that m A 2 p E I~; also aA2p <_ p 2
andpcIi ~implyaA2pCI~,mA2b4 mandmcIi ~ i m p l y m A 2 b C I ~ . Hence t ~ I~ and thus I* n F ~r ;~, which is impossible, and, consequently, I* - (I~, I~) is the required prime GI. V] C o r o l l a r y 6.3.6. Let I - (I1,/2) be a GI of a GBA A - {A~, A1, V1, Pl, ~, 4 , e, A~, A~, V2, ~ } and a E A~ \ I~. Then there exists a prime GI I* - (I~, I~ ) such that I <_ I* and a-r I [ . P r o @ Without loss of generality let a ~ A2 \ I~ and let us consider the GF aF(R) -- ( F 1 - - { x ~ A l "
a 4 x } , Fg, - {y E A2 " a <_ y } ) .
Then I n a~(~) -- ~ and it remains to use Theorem 6.3.5.
2
6.3. Generalized Filters and their Variety. . . .
267
C o r o l l a r y 6.3.7. Let A - {A1, A1, VI,~I, 1~),~ , e , A ~ , A 2 , V~, ~ } be a GBA, a, b ~ A1 ~ A2 and a r b. Then there exists a prime GI which contains exactly one of the elements a and b. Proof. Let us consider the following cases: (1) a, b ~ A~, a r b. Then for a, b E A1 (a, b c A2) and a <1 b (a < b), it
suffices to consider the GI aZ(L) and the GF bF(c) (respectively, the GI ai(R) and the GF bF(R)) since ai(c) ~ bF(c) -- ~ (ai(R) C~br(R) -- 2~). (2) a, b ~ A~, a r b and a, b are not comparable by 4 IA, = < - If, for example,
i
a, b ~ As, then we also have ai(~) ~ b~(R) - 2~ - a~(n) N hi(n). Indeed, if we consider a <_
a 4
2
and
({:;c C AI" :;c 4 b}, {y c A2" y < b}), 2 then aF(R) A bl(R) r 2~ implies t h a t there exists an element y c A2 such t h a t a _< y _< b and so a < b, which is impossible. The proof for ai(R) and bF(R) is 2 2 2 similar. (3) a E A1, b E As and a -< b. Let us consider bi(R)-
aI(L)-
({X C AI" a: < a}, {g E A2" y ~ a } ) 1
and bF(R) -- ({X ~ n l " b 4 x}, {y ~ n 2
9 b ~y}). 2
It is clear that aI(L) N bF(R) -- 2~. (4) a E A2, b c A1 and a, b are not comparable by 4 . In this case we also have aI(R)NbF(L) -- ~ -- aF(R)Nbt(L). Indeed, let, for example, aI(R)C~bF(R) r ~. Then there exists y ~ A2 such t h a t b 4 9 < a, t h a t is, b -< a, which is impossible, ffl
2
C o r o l l a r y 6.3.8. A n y GI I - (I1,12) of a GBA A - {A~, A1, V1, ~1, (~), ~ , e, A2, A2, V2, ~2} is the intersection of all prime GI 's, which contain I. Proof. Let M-
{It - (I~,I~) " I <_ It, t c T }
and I ' -
(~I~,~')I2t). tET
tCT
Let us prove t h a t I ' - I. Contrary" I' r I and so I < I'. Then there exists a E I~ \ I1 and by Corollary 6.3.6 there is a prime GI I " such that I < I " and a g I~'. But then a g I~ since I ' <_ I". D T h e o r e m 6.3.9. Let A = { A 1 , A 1 , V I , g ) I , 0 , ~ , e , A2, A2, V2, q~2} be a GBA, I = (I1,/2) be a GI and F = (FI,F2) be a GF. If I N F = (I1 N F1, I2 N F2) r ~, then {I1 to Fj, A1, V1, ~ , / 2 tO F2, A2, V2} is a G.convex G.sublattice of the G.lattice {A1, A1, V1, 0, ~ ,e, A2, A2, V2} and, conversely, any G.convex G.sublattice of {A1, A1, V1, (~, ~, e, A2, A2, V2} can be represented in a unique manner as a nonernpty intersection of a GI and a GF.
268
VI. Generalized Boolean Algebra and Related Problems
Proof. First, let I n F = (11 n F1,/2 N F2) r Q and let us prove that {I1 U F1, A1, V1, 4 , / 2 U F2, A2, Vg.} be a G.sublattice of the G.lattice {A1, A1, V1, ( 9 , 4 , e , Ag~,A2, V~.}. L e t a E ( I 1 N F 1 ) U ( h N F 2 ) a n d b E I i N F i . IfaEIjnFj, then by (1) of Definition 6.2.1, a V~ b c I~. On the other hand, a c Fj and a 4 a V i b . Hence, by (2) of Definition 6.3.1, a V i b E F l U F 2 4---> a V i b E Fi and, therefore, a Vi b E I i N Fi. Furthermore, a E Fj, b E Fi and by (1) of Definition 6.3.1, a A~ b E F~. But we also have a c Ij and a A~ b 4 a. Therefore by (2) of Definition 6.2.1, a Ai b E I~ U/2 e-----F,a Ai b c I~ and thus a A~ b E I~ n F~. Now, let us prove that the set (I1N F1)U (I2 N F2) is G.convex. We will prove only the cases, where aEIinFi,
bcIjnFj,
ccAi
and a < c 4 i
b;
the others can be proved similarly. By (2) of Definition 6.2.1, we have c E I1 U /2 .e-----F. c c Ii and by (2) of Definition 6.3.1, c c F1 U F2 .e----F. c E Fi. Thus c E Ii N Fi implies c C (11 n f l ) U (I2 n/72) so that (I1 n F1) U (I1 n F2)
is G.convex and the first part is proved. By Proposition 6.2.4, any intersection of GI's is a GI. Let us prove that this fact is also true for G.convex GI's. Suppose that {It - (I~, I~)" t c T} is a family of G.convex GI's, that is, I~ U I~ is a G.convex set for each t E T, such that
I-- (/1- n I ~ , / 2 tET
-
N tET
and a, b E I1 U 12, c 6 A1 U A2, a 4 c 4 b. Let us consider only the case, where a c I1, b c / 2 and c c A1. Then a c I~, b c I~ for each t c T and since It - (I~,It) is G.convex, c E (I~ U I t) n A 1 ,g-----5,C E I~ for each t c T. Therefore c E I1 and so c c I1 U/2. Hence the set I1 U I2 is G.convex, that is, I - (I1,12) is G.convex. Suppose that s - {L1,A1, V1, ~ ,L2, A2, V2} is a G.convex G.sublattice of { A 1 , A 1 , V 1 , O , 4 , e , A2, A2, V2}. If I - (I1,I2) and F - ( F 1 , F2) are respectively the GI and the GF, generated by (L1,L2), then Li c_ Ii n Fi. By Theorem 6.2.5 and (6) of Corollary 6.3.4, x c Ii N Fi implies that there exist elements b E Lj, c c Li such that c _< x 4 b. Therefore x E L i because L 1 U L 2 is G.convex and /
thus L i - Ii n F~. Finally, let us prove that this representation is unique. Suppose that there exist I' - (I~, I;) and F ' - (F~, F~) such that L 1 U L 2 - (I~ n f ; ) U (I; n F~). It is clear that Ii _C I ' as Li c_ I ' and I is generated by (L1,L2). Similarly, F~ C_ F'. On the other hand, let a c I ' and b E Lj be any elements. Then a V 3 b c I~ and a Vj b _> b E F~. Following (2) of Definition 6.3.1, a Vj b E F~ and, therefore, J a V j b E I } N F ~ . Furthermore, a 4 a V j b a n d L j - I~NF~ implyaVjb EIj. Therefore, by (2) of Definition 6.2.1, a 4 a Vj b implies a c Ii and thus I - I'. Similarly, one can prove that F - F ' . [2
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274
VI. Generalized Boolean Algebra and Related Problems
The rest can be proved by means of (1) of Theorem 6.1.35 in the manner as follows: if (x, y) E Ai • Ai, then y) -
h
(
j(x h
y)) -
h
(
j(x A j
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h
(y) -
h
y) (x)
Similarly, one can prove that
h~(x = ~ y) - hi(x) ~(~
.-ff~' h~(y) for (x, y) c Aj z d~.
~(~,.
(6) This statement follows immediately from Definitions 6.4.1, 6.4.6, and (1)(3) above. (7) If h - (hi, h2)" A ~ A', then by Definition 6.4.1, and (3) above, hi and G h2 are homomorphisms. Therefore the first equivalence is the well established fact. Now assume that (hl(x),h2(y)) e PA(~I) and h - ( h i , h 2 ) " ,4 ~ A' is a G G.isomorphism. Therefore hi and h2 are bijections, and (hl(x),h2(y)) e PA(~I) gives hi(x) A~ ~ ( h 2 ( y ) ) - ~t so that hi(x) A~ hl(~2(y)) - ~t and so hl(x A1 ~2(Y)) - O'. Hence x A1 P2(Y) -- O, that is, (x,y) E PA(~I). Similarly, one can prove that (h2(z), hi(v)) c PA(P~) implies (z, v) E PA(P2). Conversely, let (hl(x),h2(y)) c PA(~/1) implies (x,y) C PA(~I). We are to show t h a t hi and h2 are bijections. Let us consider only the case of h2 since the case of hi can be proved similarly. Assume that a, b c A2 and a J: b. If h2(a) - h2(b), then h2(a) A~I ~;~(h~(b)) - 0 implies h2(a) A~ hi (~2(b)) - O' and so (h2(a),hl(q;2(b)) c P A ( ~ ) . Therefore (a, ~2(b)) c PA(~2) SO that hA1 ~2(b) - ~. Thus by (5) of Corollary 6.1.32, ~ ( b ) < ~2(a) ~ a _< b. In a similar manner 1 2 one can prove that b <_ a, that is, a - b. The contradiction obtained shows that 2 h~ is a bijection. It is likewise easy to see that the implication
(h2(z),hl(v)) ~ P A ( ~ ) ~
(Z,V) ~ PA(~2)
gives t h a t h l and hz are bijections. Hence it remains to show that
((hl(x),hl(y))
~
PA(~i) and (he(z),h2(v)) ~ PA(~Pl)
imply ( x , y ) e PA(~I) and (z,v) e PA(~Pz)) ~ hi and hz are bijections. We shall prove only the first case since the proof of the second one is similar. Let (h~(x),hl(y)) e P A ( ~ ) implies (x,y) e PA(~Pl). We are to show t h a t hi is a bijection. If a, b e A1, a ~ b and h~(a) - h~ (b), then
hi(a) Ai ~ i ( h l ( b ) ) - Q}t implies hi(a) A~ h l ( ~ l ( b ) ) - Q)t and so (hi(a), hi (~1 (b)) ~ PA (1/)~). Hence (a, ~1 (b)) E PA (~1) so that hA1 ~1 ( b ) - ~}. Therefore by (12) of Corollary 6.1.33, a <_ b. Similarly, one can prove that b <_ a, 1 1 t h a t is, a - b. The obtained contradiction shows that hi is a bijection. By a similar reasoning one can prove that h~ is a bijection.
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276
VI. Generalized Boolean Algebra and Related Problems
is a GI of A.
Now let h" A ~ A' and I' - (I{, I;) be a GI of A'. G
(1) Suppose that x
hi(x) Vri hi(y)
9h l l ( I ~ ) U h~l(I~) and y
9h~-l(I~). If x
9h~-l(I~), then
9I~ and h ~ l ( h i ( x ) Vti hi(y)) - h ~ l ( h i ( x Vi y)) C hi-l(I~).
Therefore x Vi y 9h ~ l ( h i ( x Vi y)) c hi-l(I~). If x 9h~-l(Ij), then hj(x)V~ hi(y) 9I~ and, therefore,
h ~ l ( h j ( x ) Vti hi(y)) - h ~ l ( h i ( x Vi y)) c hi-l(I~). Thus x Vi y (2) If x
9h ~ l ( h i ( x Vi y)) c hi-l(I~). 9 h~-l(I~), y 9Ai and y <_ x, then hi(x)
hi(y) <_' hi(x) so that hi(y) i
y If x
9hi-l(/~), y
9h ; l ( h j ( y ) )
9 A~and
9I~ C I~ U I~. Hence
9Aj and y 4 x, then y Aj ~i(x)
Since hi(x)
Let h
(~. Therefore
9 I~,we obtain hj(y)
C hjl(z~) c hll(/i)U
C o r o l l a r y 6.4.10. I - (Ii,h)
9 I~,hi(y)
9h~l(hi(y)) C hi-l(I~) C h l l ( I ~ ) U h2-1(Is
so that h j ( y ) 4 ' h i ( x ) . y
i
9Ij c I~ U g and thus
h21(/;).
9 A ~ A ~be a G.isomorphism of GBA's. G
[-1
Then
is a GI of A if and only i f h ( I ) - (hl(I1),h2(I2)) is a GI of A'.
C o r o l l a r y 6.4.11. The image of a GF under a G.isomorhism is a G F and the inverse image of a GF under a G.homomorphism is a GF. Thus, if h : A ~ A' is a G.isornorphism, then F = (F1, F2) is a GF of A if G
and only if h(F) = (h~(F1),h2(F2)) is a GF of A'. Proof. The proof is an immediate consequence of the G.duality. 6.5.
(i,j)-Atoms
D
and Pairwise Atomic Generalized Boolean Algebras
Since every pair from PA(~I)[-J PA(~2) is supposed to play the role of zero element of a GBA A, we can extend the notion of an atom to the case of a GBA, that is, to say, we can define the notion of an atom not as an element, but as a pair of elements of a GBA. Moreover, we shall establish a connection between an atom, defined as an element and an atom, defined as a pair. D e f i n i t i o n 6.5.1. An ( i , j ) - a t o m of a GBA A - {A1, A1, Vl,q01, i~, 4 ,e, A2, A2, V2, F2} is a pair ( a , a Vj 0 ) E Ai x Aj, where a r 0 and if (c,d) E Ai x Aj, c _< a, d _< a Vj 0 , then (c, d) - (a, a Vj 0 ) or (c, d) 9PA (~i). i j It is clear that a 9Ai \ {0} ~ (a,a Vj O)-~PA(~i), and if a 9A1, then (a,a V2 O) is a (1, 2)-atom ~ (a V2 O,a) is a (2, 1)-atom.
6.5. (i, j)-Atoms and Pairwise Atomic Generalized Boolean Algebras
277
P r o p o s i t i o n 6.5.2. Let A { A 1 , A 1 , V 1 , ~ 1 , ( 9 , ~ ,e, A2, A2, V2,~2} be a GBA. Then an element a E Ai is an atom of the BA Ai - {Ai, Ai, Vi, !)i, (3, <, e} i
if and only if the pair(a, aVj(3) is an (i,j)-atom of the GBA A (3, 4 , e, A2, A~, V~, q~}.
{A1, A1, V l , ~ l ,
Pro@ Let a c Ai be an atom o f t h e BA Ai. Then by (11) of Corollary 6.1.33, aVj(~ is an atom of the BA Aj. Assume that (c, d) E Ai x Aj is a pair such that c < a and d < a Vj @. Then it is clear that (c, d) - (a, a Vj (9) or (c, d) - ((9, d) c PA (~i) J
or (c,d) - ( c , @) 6 PA(P~) or (c, d) - ((9, (9) 6 PA ( ~ ) . Conversely, let (a,a Vj (9) be an ( i , j ) - a t o m of the GBA A. If a is not an atom of the BA Ai, then there exists an element c E Ai \ {(9} such that c < a. Then the pair (c, c Vj (9) satisfies the conditions:
(c, c Vj (9)-~ PA (p~).
c < a, c Vj (9 < a Vj (9 and i j D
P r o p o s i t i o n 6.5.3. Let c a t - {A1,A1,V1,~l,(9,~ ,e, A2, A2, Vg, p2} be a GBA and (a, a V 9 (3) ~ A~ • Aj. Then (a, a Vj (3) is an (i, j)-atom if and only if for every pair (c, d) E Ai x Aj, we have
a <_ c, a Vj (9 <_ d or (a Ai c, (a Vj (9) Aj d) C PA(9~i)-
i
j
Proof. First, we assume that (a, a Vj (9) is an ( i , j ) - a t o m and (c, d) E Ai x Aj is any pair. Then for the pair (a Ai c, (a Vj (9) Aj d) it is obvious that a Ai c < a, (a Vj (9) Aj d <_ a Vj (9 i
j
and hence
a Ai c -- a, (a Vj (9) Aj d - a Vj (9, that is a
aVj@
or (a Ai c, (a Vj (9) Ajd) 6PA(qDi)
since (a, a Vj (9) is an (i, j)-atom. Conversely, suppose that (a, a Vj O) E A~ x Aj and a pair (c, d) E A~ x Aj satisfies the conditions: c _< a, d <_ a Vj (9. If a <_ c and a Vj O < d, then a - c, i
j
i
j
a V j O - d. If (a Ai c, (a Vj @) Aj d) e PA(~i), then (c,d) c PA(~i) and thus (a, a My (~1) is an (i, j)-atom. D T h e o r e m 6 . 5 . 4 . For a G B A , 4 - {A1, A1, VI, ~I, @, 4 , e, A2, A2, V2, qp2} and an element a E A1, we have the equivalences"
ai(L)--(Ii--{xEAl <
" x<_a},I2-{yEA2 1
;" ( ~ l ( a ) V 1 (9,991(a))
z,: > ~ I ( a ) F ( R ) - ( F I - { x f f A is a G. ultrafilter.
" y 6 a}) is a prime GI,'
is a (1, 2)-atom
1 9~ l ( a ) ~ x } , f 2 - { y E A 2 "
<
',
~l(a)<_y}) 2
>
278
VI. Generalized Boolean Algebra and Related Problems
Pro@ By (3) of Corollary 6.3.4, it suffices to prove only the first equivalence. First, we assume that ai(L) is a left principal GI and (qDl(a) V1 QI, qo1(a)) is a (1, 2)-atom. We have to show that ai(r) is a prime GI, that is, at(c) is a maximal GI so that aI(L) is proper and has no property to be contained in a proper GI of .4. If aI(L) is nonproper, then a - e so that (~l(a) V1 l~, qPl(a)) -- ((~, Q)), which is impossible. Now we assume that there exists a proper GI I' - (I[,I~) such that de(L) < I ~. Then there exists an element b c I[ \ I1, and, therefore, b g I 1 , that is, b _< a is false. But in that case b _< ~)1 (a) and b :/= 2/)1(a) since otherwise 1
1
a, @l(a) E I [ , and so I' - (I~, I;) is nonproper. Thus b < ~)l(a). Consider the pair 1
(b, b V2 (9). It is clear that b < ~1 (a) V 1 1~, bve ~ < ~1 (a) and (b, b V2 O) g PA (~1), 1
2
which contradicts the fact that (~l(a) V1 i~, ~ l ( a ) ) is a (1,2)-atom. Hence aI(L) is a maximal, that is, a prime GI. Conversely, assume that de(L) is a prime GI and (qpl(a) V1 Q), ~ l ( a ) ) is not a (1, 2)-atom. Then there exists a pair (c,d) ~ A1 x A2 such that c < qPl(a)V1 (~), d ~ ~ l ( a ) , 1 2
(c,d) 7~ ( ~ l ( a ) V 1 (~), qDl(a))
and (c, d) c E PA (~1)- Clearly, C < ~ l ( a ) V1 O, 1
d < qpl(a) a n d 2
(c,d) # ( ~ l ( a ) V 1 0 , ~ 9 1 ( a ) )
give C ~ @l(a) V1 O, d < @l(a) or c < @l(a) V l O, d ~ ~ l ( a ) 1
1
2
or both c < ~ l ( a ) V 1 (~) and d < ~l(a). 1
2
Without loss of generality consider the case c <_ ~ l ( a ) V 1 (~, d <2 p l ( a ) . d < ~l(a) ~ 2
a < ~2(d) and so p 2 ( d ) g I 1 . 1
1
Then
On the other hand, if d E I2, then
d 4 a, that is, d < a V2 (9 - ~(~1(a) so that 2
(~92(~1 (a)) < ~2(d) ~ 1
~)l(a) <_ qD2(d). 1
Hence we conclude that a < p2(d) and ~)l(a) <1 p2(d), that is a V1 ~ l ( a ) -- e -- ~2(d) 1
and hence d - O. But this conclusion contradicts the fact that (c,d)-~ P A ( ~ l ) Thus d g I2 and, therefore, aI(L) is not a prime GI. D T h e o r e m 6.5.5. Let A - { A 1 , A 1 , V l , ~ l , O , 4 ,e, A2,A2, V2,~2} be a GBA and (a, a V j O ) E A~ x Aj. Then (a, a V j O ) is an (i,j)-atom ~ aGF F = (F1,F2), generated by the pair ({a}, {a Vj 0}), is a G.ultrafilter ~ a GI I (~2(F2), pl(F1)), generated by the pair (~i(a) Vi O, ~i(a)), is prime.
Pro@ By the G.duality and (3) of Corollary 6.3.4, it suffices to prove only the first equivalence. Assume that (a, a Vj (9) c Ai x Aj is an ( i , j ) - a t o m and F - ( F 1 , F 2 ) , where F~ - {x ~ A~ 9 a<_ x}, Fj - {y ~ Aj 9 aVj (~ <_ y} is a GF, generated by the i j
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d:;,
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280
VI. Generalized Boolean Algebra and Related Problems
Thus, as in the first case, a Vj (~ - (~ and so x c Fj or x ~ c Fi (or both x E Fj and x' c Fi). Furthermore, we are to prove that the GF (F1, F2) is proper. Indeed, if there exists an element x c A~ such that x c F~ and ~ ( z ) c Fj, then simultaneously, a _< x and a 4 ~ ( x ) so that a - O, which is impossible since (a, a Vj O) is an i
(i, j)-atom. To prove the converse, assume that F - (F1, F2) is the G.ultrafilter, generated by the pair ({a}, {a Vj 0}) so that Fi - {x c Ai" a <_ x} and Fj - {y c Aj " a Vj I~ ~ y}. i
2
Let us prove that (a, a Vj (~) is an (i,j)-atom. If (c, d) c A~ x Aj is any pair, then c c Fi or ~i(c) E Fj(~=~ r c Fi), but not both since F - (F1,F2) is a G.ultrafilter. Similarly, d E Fj or ~j(d) E Fi(e==~ r E Fj), but not both. Hence there arise the following cases" (1) c E F ~ a n d d E F j . Thena_
i
d) - (a, e) e PA
(a
(3) a Vj 0 <_ p~(c), a Vj 0 _< d, and, therefore, J J
(a
(a Vj e)
d) - (e, a
e)
9
Thus the sufficient condition of Proposition 6.5.3 holds and so (e, a Vj ~) is an (i, j)-atom. D D e f i n i t i o n 6.5.6. A GBA A - {A1, A1, V1, ~91, I~, 4 , e, A2, A2, V2, ~2} is said to be (i,j)-atomic if each pair (c, d ) C P A ( ~ i ) contains an ( i , j ) - a t o m so that for each pair (c, d) g PA ( ~ ) , there exists an (i, j ) - a t o m (a, a Vj (~) such that a <_ c and i
aVj(~<_d. J It is clear that for a GBA A, we have: A is p-atomic
~
A is (1,2)-atomic
~
A is (2, 1)-atomic.
Also, a GBA A - {A1, A1, V1, ~1, t~, 4 , e, A2, A2, V2, ~ 2 } is p-atomic ~ BA's Ai - {Ai, Ai, Vl, ~1, O, <_ e} are atomic. i
the
Indeed, if A is p-atomic and a E A~ \ {0} is any element, then the pair (a, a Vj (~) g PA ( ~ ) and, therefore, there exists an (i, j ) - a t o m (c, c Vj (~) such that c _< a and c Vj ~ _< a Vj (~. By Proposition 6.5.2, c is an atom in the usual sense i j and thus Ai are atomic. Conversely, let A~ be atomic and (a,b) ~ (Ai x A j ) \ P A ( ~ ) S O that a r (~ r b and b <_ ~i(a) is false. Then b Ai a r ~ and since Ai are atomic, there exists J an a t o m c c A~ such that c_< b A i a . T h e r e f o r e c 4 b, t h a t i s , c V j ( ~ <_ b a n d i j
6.5. (i, j ) - A t o m s and Pairwise Atomic Generalized Boolean Algebras
281
c <_ a. Again applying Proposition 6.5.2, we conclude that (c, cVj@) is the required i
(i, j)-atom. T h e o r e m 6.5.7. A GBA A {A1,AI,VI,~I,O,~ ,e, A2, A2, V2,p2} is p-atomic if and only if( Vi at, Vj bt) E Pv(~i), where {(at, bt)}tET is a family of tET1
tET2
all ( i, j ) -atoms of .4 and T1 U T 2 - T. Proof. First, assume that A is p-atomic and ( Vi at, Vj bt) -g Pv(~i). If V, at - a, t6T1
t6T2
tET1
Vj b t - b, then by ( 6 ) o f Corollary 6.1.32, (Fj(b), F i ( a ) ) C P A ( P i ) and, therefore, t6T2
there exists an ( i , j ) - a t o m (at,,, at,, Vj O) such that at,, < pj(b), at,, Vj 0 <_ r i j since A is (i, j)-atomic. Clearly, to c T - T1 U T2 implies to c T1 or to c T2. Without loss of generality assume that to c T1. Then at(, <_ a. On the other hand, ato Vj 0 <_ q~i(a) <_ ~i(ato) i j j and so (ato, ato Vj @) c PA (P~). This result contradicts the fact that (ato, ato Vj 0 ) is an (i, j)-atom. To prove the converse, let ( Vi at, Vj bt) c Pv(Pi), where T1 U T2 - T. We t6T1
t6T2
shall prove that A is p-atomic. Let us assume the contrary, that is, there exists a pair (c, d ) E PA(P~) such that for each ( i , j ) - a t o m (at, at Vj @), we have: at <_ c is i false or at V j @ <_ d is false. Let J
T1-
{t c T " at _< c is false} and T2 - {t c T " at Vj O <_ d is false}. i j
It is obvious that T1 U T 2 - T. Moreover, by Definition 6.5.1, we have
(at Ai c, at Vj O) 6 PA(~/) for each t C T1 and
(at, (at Vj (9) Aj d) c PA (~i) for each t C T2. Therefore
at Vj 0 ~_ ~i(at) Vj ~i(c) for each t c T1 J and
at <_ ~j(at Vj (9) V~ pj(d) for each t c T2. i
Hence
at Vj 0 _~ qpi(c) for each t c T1 and at <_ pj(d) for each t c T2 j i since
(at Vj O) Aj pi(at) -- O and at Ai ~j(at Vj 0 ) -- 0 for each t c T. Clearly,
V ~ at <_ ~j(d), i
t6T2
~ t6T1
(at Vj @) _< ~i(c) j
282
VI. G e n e r a l i z e d B o o l e a n A l g e b r a a n d R e l a t e d P r o b l e m s
and if V~ at - a, Vj bt - b, t h e n tET2
tET1
a <_ ~ j ( d ) ,
b <_ ~ ( c ) j
and (a, b)
9Pv(p~)
since T2 t2 T1 - T. Therefore
~ i ( ~ j ( d ) ) <_ ~i(a) <_ b <_ ~i(c), J
J
t h a t is d <_ ~i(c)
J
J
and thus (~j(d), ~i(c)) 9P v ( ~ i ) . But in t h a t case, by (6) of Corollary 6.1.32, we have (c, d) 9P~ ( ~ ) , which is impossible. Hence ,4 is p-atomic.
6.6. G e n e r a l i z e d B o o l e a n Factor A l g e b r a s We shall define a factor algebra of a G B A A = {A1, A1, V1, ~1, ~=), ~ , e, A2, A2, V2, p2}. Let I = (I1,/2) be any proper GI of A and F = ( ~ 2 ( 2 2 ) , ~ 1 ( / 1 ) ) b e the corresponding GF. If E~ = A~ U / j t2 ~i(I~), t h e n Ai • / j = { 0 } , Ai • ~i(I~) = {e} since A1 V1A2 = {(~, e}, and Ij A ~i(Ii) = f~J since I = (I1,/2) is a proper a I .
P r o p o s i t i o n 6 . 6 . 1 . Let A = {A1, A1, V1, ~1, (~, 4 , e, A2, A2, V2, ~2} be a G B A and ~ be binary relation on the set Ei, defined in the m a n n e r as follows: i
(1) I f x, y C Ai, then x~y
,z----5, ( x - y ,
i
.,
:. ( ( ~ -
JG
ja
y ) v~ e
y-xEZj)
; :,
( y :- ~) v~ e 3(;
= y - x e I~). i
j~
= x - y, i
(2) / f x c A~, y c Ij U ~ i ( I i ) , then i ~:
i
iG
:. ( ( ~ - y ) v j e) = ~ - y e I j , ic;
(3) I f x, y
J
JG ( y - ~ ) v~ e Jc;
:
y - x e I~). i
9Ij U ~i(Ii), then x~y
i
~.
( x - y, y - x c I~) ~: :. i(;
~: :~ ( ( x - y) v j e = it;
iG
9 - y, ( y - x ) v~ e = y J
iC
x
9I j ) .
J
Then ~ is an equivalence relation on Ei such that elements f r o m Ij and y)i(Ii) i
belong to different equivalence classes. Moreover, if x, y 9Ij or x, y 9~i(Ii), then x ~ y ,z---5, x ~ y. i j Proof. First, let us prove t h a t ~ is an equivalence relation on Ei. It is obvious i
t h a t x x, x for each x i
9E~ since 11 N / 2 = {(~}. Also, by (2), the implication
x ~ y ---5, y ~ x is clear for each x, y i
i
9Ei. Therefore it remains to show x ~ y i
and y ~ z imply x ~ z. We shall consider the following cases: i
x, y, z
9Ai;
x
9A~, y, z
9Ij U ~i(Ii);
x, y
9Ai,
z
9Ij [2 ~i(I~)
6.6. G e n e r a l i z e d B o o l e a n F a c t o r A l g e b r a s
283
and
x, z ~ A~,
y ~ Ij U ~ i ( I i ) .
T h e r e s t r i c t i o n ~ IA, is an e q u i v a l e n c e r e l a t i o n on Ai since Ii is an ideal in t h e usual sense. H e n c e t h e case x, y, z ~ Ai is obvious. Since all t h e o t h e r cases are p r o v e d similarly, we shall consider only t h e case w h e r e x ~ A~ a n d y, z ~ I j U ~ ( I ~ ) . B y (2) a n d (3), we o b t a i n
x,~y i
,,t---5, (x - y ~ Ii, y - x ~ I j ) i(;
j(;
and y~
i
z ,,+---5, ( y - z , i(;
z - y c Ii) i(;
so t h a t
a~ ~j(y) c I~, y Aj ~(~) c Ij, y a~ ~j(z), ~ a~ ~j(y) e I~. T h e r e f o r e G D L 1 , G D L 2 a n d Definition 6.2.1 i m p l y
(x A~ ~j(y)) v~ (y A~ ~j(~))- ((~ A~ ~j(y)) vj y) A~ ((x A~ ~j(y)) v~ ~j(z))(~ v~ y) A~ (~j(y) vj y) A~ (~ v~ ~j(~)) A~ (~j(y) v~ ~j(~)) = (x vj y) A~ (~ v~ ~(~)) A~ ( ~ ( y ) v~ ~j(~)) 9I~ since I = ( I 1 , / 2 ) is a GI. Clearly, i
Following (2) of Definition 6.2.1, x Ai ~ j ( z ) = x - z e Ii. i(;
In a similar m a n n e r one can show t h a t z - y E Ii a n d y i (;
J c;
x E Ij i m p l y
Z--XCIj.
H e n c e x ~ z and, therefore, ~ is an e q u i v a l e n c e r e l a t i o n on Ei. i
i
F u r t h e r m o r e , let us prove t h a t e l e m e n t s from Ij a n d ~ i ( I i ) b e l o n g to different e q u i v a l e n c e classes. If x c Ij, y c y:i(I~) a n d x ~ y, t h e n t h e r e is an e l e m e n t z E I~ i
such t h a t y = y)i(z) as y c qpi(Ii). Moreover, x ~ y implies x Ai ~ i ( Y ) , Y Ai ~ j ( x ) C Ii, i t h a t is, a n d so z Vj x c ~ ( I ~ ) . O n t h e o t h e r h a n d , z ~ I~, x ~ I o i m p l y z Vj x ~ Ij a n d h e n c e z Vj x ~ Ij N ~ i ( I i ) , which is impossible since I 3 A p i ( I i ) = ~ . Finally, let x, y ~ I o or x, y ~ ~ ( L ) . It is obvious t h a t Ij U ~ ( I ~ ) C E~ C~A j . T h e r e f o r e , on t h e one h a n d , by (3), for a pair x, y ~ Ij U ~ i ( I i ) c Ei, we have x ~ y ~ (x - y, y - x E I~), and, on t h e o t h e r h a n d , by (1), for t h e s a m e pair i
i(,
i(;
x, y ~ Ij U ~ i ( I i ) C Ei, we h a v e x ~ y .z----5. (x - y, y - x ~ I~). T h u s , if x, y ~ Ij j
or x, y C ~ ( I ~ ) , t h e n x ~ y i
i(;
i(;
.z--5. x ~ y. j
Let us d e n o t e t h e set of all ~ - e q u i v a l e n c e classes by E i / I ,
E{/I - {[x] {- x s E{}.
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288
VI. Generalized Boolean Algebra and Related P r o b l e m s
Thus the algebra . A / I - {El/I, [-]1, l-]1, (1)1, [l~], ~ F, [e], E2/I, R2,112, ~2} is a G B A t h a t we call a G.Boolean factor algebra of the G B A A - {A1,A2, Vl, ~1, (9, 4 , e, A2, A2, V2, ~2}. Moreover, if we consider the pair n - (nl, n2), where the maps ni" Ai ~ E i / I are defined as ni(x) - [x] i, then it is easy to see t h a t n ( r t l , rt2) " fl[ ---+ .A/I. This G . h o m o m o r p h i s m is called a natural G.homomorphism. G
Finally, let us consider any G . h o m o m o r p h i s m h - (hi, h2)
.4 9 ~ A ~. Then, by G
Example 6.4.2, h - 1 ( O ') - (h[l((~'),h~l((~')) is a proper GI of .4. Let A / h - I ( O ') be the G.Boolean factor algebra. If g-(g1, g2), where gi" Ei/h-l((~') ~ A~ are de9 ~a .4 ~ is a G.isomorphism. fined as gi([x] i) - hi(x) , then g - (91,92) A/h-l(~)') Indeed, by (7) of T h e o r e m 6.4.7, it suffices to prove only t h a t gl([X] 1) - (9' and g2([y] 2) - (9' imply [x] 1 - [(9] and [y]2 _ [(9]. But this implication is an immediate consequence of the definition of g - ( g l , 9 2 ) . 6.7. Generalized Fields of Sets and the Generalized Field Representation of a Generalized Boolean Algebra D e f i n i t i o n 6.7.1. Let X be a n o n e m p t y set, Ai(X) be a ring of subsets of the set X , t h a t is, Ai(X) be closed under the finite set-theoretic operations intersection and union, ==<: be a quasi order relation on A I ( X ) u A2(X) such t h a t ==~ IA,(x) = C , A I ( X ) N A 2 ( X ) = {2~, X } and A c Ai(X) implies X \ A c Aj(X). If there exists maps []i: Ai(X) ~ A j ( X ) satisfying the conditions:
(1) A ~ [A]~, [ [A]~]j = A and X \ [A]~ = IX \ A]j for each set A e A~(X). (2) [ ] i o n = No([ ]i, []i) a n d [ ] i o U = Uo([ ]i, []i) s o t h a t [ANB]i = [AIiA[B]i and [A U B]i = [A]i U [B]/ for each pair (A, B) c A i ( X ) x A i ( X ) , then we shall say t h a t Jc(X) = {AI(X)UA2(X),N, u, \, ==<:, []1, []2]} is a G.field of sets (briefly, GFS) for which X is the basic set. Clearly, []1 and []2 are bijections and []i - - [ ] ; 1 It is easy to verify t h a t the equivalence (A =<: [A]i and [ [A]i]j - A) ,z---->,A ==~ [A]i ==<: A holds for each set A E Ai(X). Indeed, A :=<: [A]i and [[A]i]j - A imply [A]i =:<: [[A]i]j - A. On the other hand, if A ==<: [A]i =<: A, then A =:< [A]~ ==< [ [A]/]j =:< [A]~ =:< A and so A ~ [[A]~]j ==<: A. Therefore A c_ [[A]~]j c_ A since ==<: latex) and thus [ [AJ~]j - A. Obviously [2~]i- o and [ X ] i - X. Moreover, for each set A e Ai(X), the set [A]i is a smallest set from A j ( X ) (with respect to the set-theoretic operation inclusion) such t h a t A ==~ [A]~. Indeed, if B c ,Aj(X) is any set such t h a t A ==~ B C [A]i, then [A]i ==<: A ==<: B c [A]i, t h a t is [A]i c_ B c_ [A]~ and thus B -
[A]~.
6.7. G e n e r a l i z e d F i e l d s of Sets a n d . . .
289
For a GFS ~'(X) - { A I ( X ) U . A 2 ( X ) , N , U, \, ==<, []1, []2} let us define binary operations ni, Ui " (A~(X) x A i ( X ) ) u ( A j ( X ) x A i ( x ) ) --+ A i ( X ) as follows: AN~B-ANB,
AU~B-AUB,
if ( A , B ) r ( A i ( X ) • A i ( X ) ) \ {(A, [X \ A]j) " A r A~(X)}, and A Ni [X \ A]j - ~, AN~ B - [A]j N~ B,
A Ui [X \ A]j - X if A c A ~ ( X ) ;
A U~ B - [A]j U~ B if ( A , B ) e A j ( X ) x A~(X)
so that ANiB--
~'
if A - X \ B , if A ~ X \ B ,
( [A]jNB,
A U i B - ~X'
if A - X \ B , if A C X \ B .
[ [A]jUB,
Hence for ( A , B ) c A j ( X ) x A~(X), we have A n~ B C_ B,
A o~ B c_ [A]j ~
A and B c_ A U~ B,
A ~
[A]j c_ A U~ B.
From the above arguments, we immediately conclude that [A n~ Bli - [A]i nj [B]i,
[A Ui B]~ - [ A ] i
Uj
[B]i if (A, B) r A i ( X ) x A i ( X )
and [A N, B]~ - [ IA]j N~ B]i - A Nj [B]~, [ A U ~ B ] i - [ [ A ] j U ~ B ] i - A U j I B I i if (A,B) E A j ( X ) x A i ( X ) . Hence it is not difficult to see that every GFS can be treated as a GBA A(~'(X)) with ~ instead of 4 , c instead of <, ~ and X instead of 0 and e, i
respectively, the set-theoretic complementation \ instead of Wi, []~ instead of X , Oi and U~ instead of Ai and Vi, respectively. (The converse of this statement is Theorem 6.7.5.) Indeed, let us first consider the question of coordination of the quasi order - - < and the lattice operations ni, Ui to show that A ==< B .e---->.A Ui B - B for each pair ( A , B ) c ( A i ( X ) x A i ( X ) ) u ( A j ( X ) x A~(X)). If (A, B) c A ~ ( X ) ( x A ~ ( X ) , then A=:< B .<-->. A C_ B ..e->. B - A U B ..~->. B - A U~ B, and if ( A , B ) c A j ( X ) x A i ( X ) , then A ::::< B ~
[A]j c_ B ..v-->. B - [A]j u B ~
B - [A]j u~ B ~
B - A u~ B.
Now the examination of axioms L1, L2, GL3, GL4, GDL1 and GDL2 reduces to simple calculations which readily show that A(.T'(X)) - {AI(X),N1, Ul, \ , ~ , --<:, X, A i ( X ) , N2, U2, \} is a GBA. A GFS 5r(X) - {.41(X) U A2(X), N, U, \, ==<, []1, []2} is said to be reduced if for every pair of distinct points x, y c X there exists a set A c A i ( X ) such that x E A and y - c A .
290
VI. Generalized Boolean Algebra and Related Problems
D e f i n i t i o n 6.7.2. A G.field representation of a GBA .4 - {A1, A1, V l , 991, (~), 4 , e, A2, As, V2,992} is a pair ( f , X ) , where X is the basic set of a GFS Jc(X) considered as a GBA A(Jz(X)) = {A1(X),cll,U1, \,;g, ===~,X,A~(X), Ns, Us, \}, and f 9 A ~ A ( ~ ( X ) ) is a G.isomorphism such that f - (fl, f2), G
where f ~ ' A ~ ~ A~(X) are bijections, and the following conditions hold: (1)
s o Ai -- rli o ( f i , s
and s o Vi -
Ui o ( f i , f i )
so that
f~(~ A~ y) - f~(x) n~ s
~nd f~(~ v~ y) - s
u~ f~(y)
for each pair (x, y) c Ai x Ai. (2)
f i o Ai -- rli o
(fj, fi)
a n d f~ o Vi - Ui o
(fj, fi)
so that f~(x/~
y) - f j ( x ) n ~
s
y)
f~(y) - [ f j ( x ) ] j
n~ f~(y)
and v~
=
f~(x) u~ s
=
[f;(x)]j u~ f~(y)
for each pair (x, y) E Aj x Ai. (3) \ o fi - fj o 99~ so t h a t X \ fi(x) = fj(99i(x)) for each x 6 A~. Note t h a t here we also identify these (reduced) G.field representations (f, X) and ( I ' , X ' ) of a GBA ,4 = {A1,A1,V1,991, O , ~ ,e, As, As, V2,992} which are equivalent in the sense as follows: there exists a bijection h : X ~ X t and a G.isomorphism h* = (h~,h~) : A ( 5 ( X ) ) ~ A ( ~ ( X ' ) ) such that h~(A) = {h(x) : x c A} for every set A c X i ( X ) . L e m m a 6.7.3. If (f , X ) is a G.field representation of a GBA .4 = {A1, A1, V1, 991,O, ~ ,e, A2, As, V2,992}, then for each x c Aj we have [fj(x)]j = f~(x V~ 0).
Proof. By (6) of Theorem 6.1.16, x c Aj implies x V~ 0 6 x and, therefore, by (1) of Definition 6.7.1 and the G.isotonicity of f = (fl, f2), we have f i ( x Vi (9)==3
[fj(x)]j,
that is fi(x Vi (~) c_
[fj(x)]j.
On the other hand, the same arguments give fj(x)==~ fi(x Vi (9) and, hence, by the definition of the set [fj(x)]j, we have fj(x)==< [fj(x)]j c_ fi(x Vi (9). Thus
[fj(x)]j - f i ( x Vi 0).
[3
We are now ready for the principal result of this section; namely, the next theorem characterizes all the reduced G.field representations of a GBA .4. The existence of a reduced G.field representation will be a consequence of this theorem and of the results of previous sections. T h e o r e m 6.7.4. There exists a one-to-one correspondence between the reduced G.field representations of a GBA A and the Stone families of prime GI's of X.
Pro@ Let S be a Stone family of prime GI's of a GBA A. The reduced G.field representation (f, X) of A can be associated with S in the manner as follows: let
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292
VI. Generalized Boolean Algebra and Related P r o b l e m s
Conversely, if x1 ~ f j ( a ) A~ f~(b) - [fj(a)]j ~ k ( b ) , then
f j ( a ) Ni fi(b) - [fj(a)]j N f~(b) so t h a t x1 ~ [fj(a)]j ~ f~(b) a n d s o x i ~ f ~ ( a V ~ ) ~ f ~ ( b ) . Therefore a V i @ g I ~ z--->, a - g I j and bgI~. Now, Definition 6.2.9 gives aA~b-gIi so t h a t x i e f~(aA~b) and thus f~(aA~b) - f j ( a ) ~ f ~ ( b ) . Furthermore, if x~ ~ f~(a V~ b), where a ~ A j , b ~ Ai, then a V~ b--gI~. By (1) of Definition 6.2.1 the case a ~ Ij, b ~ I~ is impossible. Hence a-g Ij or b g I~ (or both a g Ij and b g/~). We shall consider only the cases a g Ij, b ~ /~ or a g Ij,
bg~. If a-g I j, b ~ I~ and b -
~ j ( a ) , then
f j ( a ) ~ fi(b) - [fj(a)]j ~2~ f~(b) - [fj(a)]j t3~ f i ( ~ j ( a ) ) - [fj(a)]j ~ ( X \ f j ( a ) ) - X so t h a t x i ~ f j ( a ) ~ fi(b). Now we assume t h a t b --/: ~j(a). Then
f j ( a ) tAi fi(b) - [fj(a)]j Ui fi(b) - [fj(a)]j ~ fi(b) and, therefore, g-EIj ~
g V i {~--EI i ,g------5,xI E f i ( g V{ ~ ) -- [fj(g)]j,
t h a t is, x1 ~ f j(a) Ui k ( b ) . I f a - g I j , b-gIi, then, by (2)of Theorem 6.2.10, the case b - ~ j ( a ) is impossible. Hence f j ( a ) U i fi(b) - [fj(a)]j U fi(b) and a - g I j , b-gIi imply a V~ O - g I i , b-gIi so that x1 e k ( a Vi O ) U fi(b) - [fj(a)]j U fi(b) - f j ( a ) Ui fi(b). Conversely, let x , c f j ( a ) Ui fi(b) and b - ~j(a). Then
f j ( a ) t& fi(b) - [fj(a)]j tAi f i ( ~ j ( a ) ) - [fj(a)]j tAi ( X \ f j ( a ) ) - X and, by (2) of Theorem 6.2.10, a-EIj or b - qpj(a)-EIi. It is clear t h a t always a V~ b-~Ii so t h a t x1 c f i ( a Vi b). Now let b 7~ ~j(a). Then f j ( a ) Ui f~(b) [fj(a)]jU fi(b) and so x1 E [fj(a)]j - fi(aVi(~) or x i c fi(b) (or both x i c [fj(a)]j and x i E fi(b)). Clearly, in all cases a Vi b E I i so that x i c f i ( a Vi b) and thus f i ( a Vi b) - f j ( a ) tO~ fi(b). We have shown that the conditions (1)-(3) of Definition 6.7.2 are fulfilled and, therefore, f - (fl, f2) " A - {A1, A1, Vl, q;1, O, 4 , e, A2, A2, V2, P2} -+ f(.A) A ( . T ' ( X ) ) - { A I ( X ) - { f l ( a ) - { x i " a-gI1 ..r a v2 ~ ) g I 2 , I - (11,12) c S } " a c A 1 } , N 1 , U I , \ , , ~ , ===<,X,.A2(X) - {f2(b) - { x I " b-gI2 ~ bV10cIl, I(11,12) E S} 9 b EA2}, C12,tO2, \} is a G.homomorphism of the GBA A onto the GFS ~ ( X ) , considered as the GBA A ( . T ( X ) ) . To prove t h a t f is a G.isomorphism, it remains to show only t h a t fl and f2 are bijections so that by virtue of (7) of Theorem 6.4.7, it suffices to show t h a t if f l ( a ) - f2(b) - ~, then a - b - O. But this implication immediately follows from the definition of f~ and (1) of Theorem 6.2.16. Furthermore, let us prove t h a t ~ ( X ) is reduced. Indeed, if x i , xi, E X and xx ~ x i,, then by the definition of the set X, we have I - (I1, I2) r I ' - (I~, I~). Therefore since I and I ' are prime GI's, there exist elements a, a' E A1, b, b' E Az such t h a t a f t I1,
a-El~,
a'-EI1,
a' E I~, b E I2, b g I ~ ,
b'gI2,
b' c i r .
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294
VI. G e n e r a l i z e d Boolean A l g e b r a a n d R e l a t e d P r o b l e m s
and thus x - ~ f j ( b ) U [fj(b)]j, that is, b E I x. Hence we conclude that Ix = (I~, I f ) is a GI. Moreover, the proof of (1) immediately implies that Ix is a proper GI. It remains to prove that Ix = (I~, I~) is a prime GI, that is, the condition of Definition 6.2.9 holds. First, assume that a, b ~ A~ and a A~ b E I x. Then
x-~ f~(a A~ b)U [f~(a A~ b)]~ - (f~(a) n f~(b)) U ([f~(a)]i A [f~(b)]~). If we assume the opposite, that is, a g I x, b g I x, then
x E (fi(a)U [f~(a)]~)N (f~(b)U [f~(b)]~).
(I)
Since a Ai b E I ,x, (I) immediately implies
x E f~(a)rq (X \ [f~(a)]~) rq (X \ f~(b)) rq [f~(b)]~
(II)
x E (X \ f~(a)) n [f~(a)]~ rq f~(b)0 (X \ [f~(b)]~).
(III)
or
By the definition of maps f~, in the case (II), we have
xE[fi(a)]~ - fj(a Vj e) ,z---->,a Vj e E I f ~
a c Ix ~
x E fi(a),
which is impossible. Similarly,
x--~ f~(b) ~
b E I~ ~
b Vj 0 E I f ~
x-E fj(b Vj 0) - [fi(b)li,
which is also impossible. Using similar arguments, one can prove that (III) is impossible. Thus if a, b E Ai and a A~ b E I~, then
x--~fi(a) U [fi(a)]i or x - c f i ( b ) U [f~(b)]i ( o r both x-E (Si(a)U [S~(a)]i) U (Si(b)U [S~(b)]~) ) and s o a c I x o r b E I x (or b o t h a c I x a n d b E I X ) . Finally, let a E Aj, b E Ai and a Ai b E I x. Then
x-E fi(a Ai b) U [fi(a Ai b)]i - (fj(a) ni fi(b)) U [fj(a) ni fi(b)]i = = ([fj(a)]j ni fi(b)) U ([fj(a)]j Nj [fi(b)]i) -
= ([fj(a)]j N fi(b)) U ( f j ( a ) N [fi(b)]i) and, using the arguments as above, we obtain a E I f or b E I x (or both a E I f and b E I / ) . Thus Ix - (I~, I f ) , where I.~ - {a c A~" x--~f~(a)U [fi(a)]~}, is a prime GI. Now let us prove that S - {Ix - (I~, I f ) " x E X } is a Stone family of prime GI's. Indeed, let a c Ai \ {fg} be any element. Then fi(a) ~ ;g and so, there exists an element x E fi(a). But by the definition of maps f~, we have a g I x and, therefore, (2) of Theorem 6.2.16 implies that S is a Stone family. Thus, we have constructed two maps Ct I and a2, where O~1 associates the reduced G.field representation of A with every Stone family S of prime GI's of .4 and vice versa: a2 associates the Stone family S of prime GI's of A with every reduced G.field representation (f, X) of ,4.
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296
VI. Generalized Boolean Algebra and Related Problems
(f', X'), then by the definition of (f', X') the points of X ' are in the one-to-one correspondence with the prime GI's of S and, accordingly, with the points of X. Moreover, following the corresponding definition,
f{(a) - { x e X " a-~I f ~ for each element a
a vj O-~I~, Ix - ( I { , I ~ )
e S}
9Ai and
e f'(a)
a-
Ix
so that f~(a) = fi(a) for every element a
a
v; e-cI
x c f
(a)
9Ai. This completes the proof.
D
By virtue of Corollary 6.2.14, the family S of all prime GI's of a GBA ,4 is a Stone family and on the strength of the preceding theorem, there exists a reduced G.field representation of.4 associated with S. Thus the following principal statement is valid. T h e o r e m 6.7.5. Every GBA is G.isomorphic to a reduced GFS.
Proof. Indeed, the family of all prime GI's of a GBA A is a Stone family. By Theorem 6.7.4, to this Stone family there corresponds the G.field representation ( f , X ) of A and from Definition 6.7.2, it follows that the GFS De(X), considered as the GBA A ( ~ ( X ) ) = {AI(X) = {fl(a) : 9A 1 } , n l , U l , \ , ~ , ==<,X, A 2 ( X ) = {f2(b) : b 9A2}, N2, u2, \}, is G.isomorphic to A. [~ C o r o l l a r y 6.7.6. For each real number m > 0 and every non-zero element a 9Ai there exists a G.quasi measure # = (p1,#2) on a GBA A = {A1, A1, V1, pl, O, ~ ,e, A2, A2, V2, ~2} such that #i(a) = # j ( a V j O) = m .
Pro@ Let m > 0 be any real number and a c A~ \ (2~} be an arbitrary element. Then, by definition, fi(a) -r ;g =/=f j ( a Vj O) and, therefore, there exists an element b 9f~(a). If we define a pair # = (#1,#2) as
{
?T~, if b e fi(x) if b-~ fi(x) for each x E Ai
-
0,
and
{77~, 'J(Y)
-
0,
if b Vj 0 c f j(y) if b Vj @-E f j (y) for each y c Aj '
then it is not difficult to see that # = (~tl, it2) is the G.quasi measure on ~4 such that #i(a) = # j ( a V j (~) -- m . D The reduced G.field representation of a GBA A, associated with the family of all prime GI's, is called the perfect G.field representation of .4. Theorem 6.7.5 concludes one part of the study of representations of a GBA. In the next section we shall investigate the second part of the modification of Stone's representation theorem.
6.8. Bitopological Representation ...
297
6.8. B i t o p o l o g i c a l R e p r e s e n t a t i o n of a G e n e r a l i z e d B o o l e a n A l g e b r a Let A = {A1,A1,VI,g~I,O, 4 , e , A2, A2, V2, q~2} b e a GBA, S = { I = (11,/2)} be any Stone family of prime GI's of A and (f, X) be the reduced G.field representation of A associated with S in the sense of the first part of Theorem 6.7.4. We can endow the set X with the topologies rl and 72 by taking the rings of sets .41(X) and A 2 ( X ) of the GFS 2r(X) = { A I ( X ) U A 2 ( X ) , N, U, \, ==<, []1, []2} as the corresponding bases of the open sets. Clearly, t h e / - o p e n sets have the base consisting of j-closed sets since A i ( X ) = co A j ( X ) and, therefore, (1) of Corollary 3.1.6 implies that p-ind X = 0. On the other hand, if we assume that x, y E X, x -r y, then/V(X) is reduced implies, in particular, that there exist A, B c A I ( X ) such that x c A, y g A and x g B, y c B. It is clear that
A, B E rl NCOr2, X \ A, X \ B E r2 DICOrl so that
U(x) = A c 7-1, U(y) = X \ A c T2, V(x) = X \ B C ~-2 and V(y) = B E T1 are disjoint neighborhoods of the points x, y E X and thus, by (9) of Definition 0.1.6, (X, T1, T2) is p-Hausdorff.
T h e o r e m 6.8.1. Let S = {I = (I1,/2)} be a Stone family of prime GI's of a GBA A and let (f, X ) be the corresponding reduced G.field representation of A. Then the BS (X, T1,7-2) is FHP-compact if and only if S contains all the prime GI's of .4, that is, if and only if (f, X ) is the perfect G.field representation of A. Proof. First, assume that S = {I = (I1,/2)} is the Stone family of all prime GI's of the GBA A. It suffices to prove that every p-open covering U C {{fl(a)
9 a cA1}, {f2(b) " b c A2}}
of X such that b/N T1 r ~ r ~ ~'17-2 has a finite subcovering. Assume the opposite, that is, there exists a p-open covering { { f l ( a n ) : rt C N } , {f2(bm) : rrt E M } } of X so that hEN
and
mEM
# ( U Sl/an/)o ( U hEN1
mEM1
for any finite subsets N1 c N and M1 c M. Therefore, alVla2V1...Vlan~e,
bl V2 b2 V2 "" V2 bm C e
and ~1(al V1 ae VI'-" V1 a~) ~ bl Ve be V e " . V2 bm is false for any finite subsets N1 = { 1 , 2 , . . . , n } C N, M1 = { 1 , 2 , . . . , m } C M. Indeed, if a l V1 a2 V1 ... V1 an ~: e, then f l (al V1 a2 V 1 . . . V1 an) 7s f(e) - X
and so
f l ( g l Vl g2 Vl "'" Vl
an)
--
f l ( g l ) Ul f l ( g 2 ) U l " ' " Ul/1(an)
-
-
298
VI. Generalized Boolean Algebra and Related Problems = fl(al) U fl(a2) U... U
since f -
(fl,f2)
al V l a 2 V 1 . . . V l a n
fl(an) 5r X,
is a G.isomorphism. The same is true for bl,b2,...,bm. If - - a , b l V2b2V2---V2bm - b and ~bl(a) 4 b, t h a t is, g91(a) < b, 2
then f - ( f l , f2) is a G.isomorphism implies t h a t q~l(a) _< b ,e---->,f2(~l(a)) C_ f2(b) ,e----->,X \ / l ( a ) c_ f2(b)<--->, fl(a) U f 2 ( b ) - X since 4
2 IA,--~
and =:< A , ( x ) - C _ .
Thus the simultaneous fulfilment of all three cases al Vl a2 V1 "'" V1 an r e, bl V2 b2 V2 "-. V2 bm r e and ~31(al V1 a2 V l " ' " Vl
an) ~ bl V2 b2 V 2 " "
V2 bm
is false enables us to conclude t h a t
X ~ (
U fl(an))U
nCN1
(
U
mCM1
f2(bm))
for any finite subsets N1 C N and M1 C M. Let I - (I1,/2) be the GI generated by the pair of sets (B1 - {an " n E N}, B2 - {bin 9 m eM}). Then I - (I1,12) is proper since the contrary means t h a t e c I1 N I2 and, by Theorem 6.2.5, there exists, in particular, a finite sequence of elements a l , a 2 , . . . ,ak E B1 such t h a t al V l a 2 V 1 - - . V l a k - e, which is impossible. Let us consider a family of all proper GI's ordered by the partial order < defined in the part preceding Proposition 6.2.4. Then, by virtue of the latter proposition, every chain in this family has the upper bound, also defined by this proposition. Therefore Zorn's lemma implies t h a t this family has a maximal element, namely, a prime GI I. - (I~,I~) such t h a t I < I.. Clearly, an c Ii ~ ~ an V2 (9 c I~ for each n E N and bm C I~ ~ bm V1 I~ E Ii ~ for each m c M. Also, I* c S. But ( f , X ) is the reduced G.field representation of the G B A A which corresponds to S and, by (.) in the proof of Theorem 6.7.4, xi. g f l ( a n ) and xI. gf2(bm) for each n c N and m c M. Thus xL g ( U fl(an)) U ( U f2(b~)) - x . This contradiction proves the first part
nCN
mCM
of the theorem. Conversely, let (X, rl, r2) be F H P - c o m p a c t and there exists a prime GI I (I1,/2) such t h a t I g S so t h a t I r Ix for each point x c X. Therefore for each point x E X, the following conditions hold: (1) There exists an element a E A1 such t h a t a c
I1 ,z----F,a V2 0 E / 2
and a-gI~ ,e----->,a V2 ( g g I ~ ,e---->,x c f l ( a )
and there exists an element b c A2 such t h a t b g / 2 <---> b V 1 O g I 1
and b E I~ ,e---->, b V 1 1~ E I~ ,e----->,x-gf2(b).
(2) There exists an element c E A1 such t h a t
c-CI1 ~
c V20g/2
and c c I~ <---> c V2 0 C I~ ,e---->x g fl(c)
6.8. Bitopological Representation ...
299
and there exists an element d c A2 such t h a t d c / 2 ,z---5, dV1 (3 c I1 and d - ~ I ~ ~ Since for each point x e also prime, in both cases (1) d as ~1 (c), that is, b = ~ l ( a ) , condition (1) above. Indeed, for each element that
X the GI Ix = ( I ~ , I ~ ) is prime and I = (I1,I2) is and (2) one can consider the element b as ~ l ( a ) and d = q~l(C). As will be seen below, it suffices to apply x c X, there exists, by (1), an element a x c A1 such
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a~ E I1 ~
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{fl (ax): x c X} is
1-op
ax V2 O ~ I~ <---->.x E fl (ax).
cov i g of X.
b c I2 \ { e }
element. Then f2(b) ~ ;g and it is obvious that {{fl(ax) : x E X}, {f2(b)}} is p-open covering of X so that, there exists its finite subcovering since (X, T1, ~-2) is FHP-compact. If this finite subcovering does not contain the set f2(b), t h a t is, if it has the form { f l ( a x k ) : xk c X, k - 1,n}, then f l ( a x l ) U f l ( a x 2 ) U . . . U f l ( a x , , ) X and so axl V1 ax~ V1 "'" V1 a~,, = a = e since, like in the proof of the first part of this theorem, for a :/: c, we obtain f l (a) ~ X~ that is, f l ( a X l V1 ax 2 V I . . . V1 ax,,) = f l ( a X l ) [-Jl f l ( a x 2 ) U l . . . [-Jl f l ( a z , , ) ~ X ,' :,
> fl(aXl)
;.
9f l ( a x 2 ) U " . U f l ( a x , , ) r X .
But ax~ c I1 for each k = 1, n and, therefore, by (1) of Definition 6.2.1, a = e c I1, which is impossible since I = ( / 1 , h ) is a prime GI. Thus f l ( a ) r X and hence finite subcovering has the form {{fl(ax~) : xk c X, k = 1,n}, {f2(b)}}. Thus f l ( a x l ) U f l ( a x 2 ) U . . . U fl(ax,~) U f2(b) - X ,', > z, ,, f l ( a ) U f2(b) = X z----->, ~ l ( a ) < b, 2
which is also impossible since a c I1, b c / 2 and I = (I1,/2) is a prime, t h a t is, a proper GI. Therefore I c S so t h a t S contains all prime GI's of the GBA A. D By virtue of the reasoning t h a t precedes Theorem 6.8.1, we see t h a t with every Stone family S of prime GI's of a GBA A one can associate a p-zero-dimensional and p-Hausdorff BS which, by Theorem 6.8.1, is F H P - c o m p a c t if and only if S contains all prime GI's of ,4. The Stone BS of the GBA A is a p-zero-dimensional, p-Hausdorff, and F H P - c o m p a c t BS, which is associated with the Stone family of all prime GI's of S. This BS will be denoted by ( S ( A ) , T1, T2). Theorem
6.8.2. Let A - {A~,A1,VI,~91,1~, ~ ,e,A~,A2, V2,~2} be a GBA, its associated Stone BS, A ( X ) = {A ~ 2 x : A = A1 A A2 or A - A1 U A2, A~ E A ~ ( X ) - 7~ A coTj} and the following conditions hold:
(S(A), T1,7-2) be
(1) the combination of l-closure and 2-interior operators and the l-closure operator as well as the combination of 2-closure and l - i n t e r i o r operators and the 2-closure operator are conjugate over A ( X ) so that ~-2 int T1 C1 A -- T1 C17-2 int T1 cl A and T1 int ~-2 cl A = ~-2 cl T1 int 7-2 cl A f o r each set A c A ( X ) .
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6.8. Bitopological Representation ...
303
Furthermore, prove t h a t A' is G.isomorphic to .4. Since (S(A),7-1,7-2) is the Stone BS of the GBA A, the pair ( f , S ( A ) ) is the perfect G.field representation of A and, by virtue of Theorem 6.7.5, the GFS .F(S(A)), considered as the GBA A(Jz(S(A))) = {fl(a) : a 6 A~},N1, Ul, \,2~, ==~ ,S(A), {f2(b) : b c A~}, N2, U2, \}, is G.isomorphic to A. Thus it remains only to prove t h a t A(.T(S(A))) is G.isomorphic to A'. First, let us show t h a t {fi(a) : a E A~} = ~-i N coTj. By the definition of ~-i, it is obvious that fi(a) c ~-i n co Tj for each element a c A~. On the other hand, let U c ~-i N co~-j be any set. Since the family {fi(a) : a E A~} is the base for 7-i, we have U = U Ut, where tET
Ut c {fi(a) :
a c A~} for each t c T. Therefore by Lemma 3 from [113], U is /-compact since U c coTj and (S(A),T1,T2) is FHP-compact. Hence the /-open covering {Ut : t c T} has a finite subcovering {Ut~: k = 1, n}, t h a t is,
U-
n
U Ut~. But Ut~ c { f i ( a ) " a c A~} for each k c 1, n and since the family
k=l
{fi(a) : a c U c {fi(a) : a the pair h = maps, then it
A~} is closed under the set-theoretic operation union, we obtain c A ~ and hence {fi(a) : a c A~} = ~-i N coTj. Now, if we consider (hi,h2), where hi : {f~(a): a c A~} -+ ~-i N coTj are the identity is obvious t h a t since the GFS's
{{fl(a)'a6
A~Iu{f2(b)'b6A~I,N,U,\,
=~,[]1,[]2}
and
{(TI n COT2)U (T2 n coT1),N,U,\ , ==x(', []i, []~ } coincide, the corresponding GBA's will coincide too. Therefore the GBA's A(~c(S(A))) and A' are the same so t h a t A' is G.isomorphic to A. []
T h e o r e m 6.8.3 (A Generalized Version of Stone's Representation Theorem). Under the hypotheses of Theorem 6.8.2, there exists a one-to-one correspondence (up to a G.isomorphism and a d-homeomorphism) between G B A ' s and p-zero-dimensional, p-Hausdorff and FHR-compact (also called Boolean) BS's such that for every GBA A the GBA
.A(.)~-(S(A)))- {T 1NCOT2, N I , U I , \ , ~ , ==~,S(A),T2 NCOTI,N2, U2,\}, corresponding to the GFS 5(S(A))
\, =<, I11, I
of the associated Boolean BS (S(M), T1, T2), is G.isomorphic to A. Proof. First, assume that A and A' are G.isomorphic. We shall show that the corresponding perfect G.field representations (f, S(A)) and (f', S(A')) are equivalent so that, there exist a bijection h : S(A) + S(A') and a G.isomorphism h* - ( h * I , h ~ ) ' A ( Y ( S ( A ) ) ) ~ A'(gc(S(A'))) such that h~(A) = {h(z) : x c A} for every set By Theorem 6.4.9 the image of a GI under a G.isomorphism preserves prime GI's as well. Ix = ( I ~ , I f ) be the corresponding prime GI.
A c {fi(a) : a c Ai} = ri n c o r j . a G.isomorphism is a GI. Clearly, Let x c S(A) be any point and I f g = (gl,g2) : A + A' is the
G
304
VI. Generalized Boolean Algebra and Related Problems
above-mentioned G.isomorphism, then g ( I x ) = ( g l ( I ~ ) , g 2 ( I ~ ) ) i s the prime GI of A ~. Let the corresponding fixed point from S ( A ~) be the point Yg(lx) and define h: S ( A ) ~ S ( A ' ) in the manner as follows: h(x) = Yg(lx). Clearly, h is a bijection. Let us prove that if B 9{fi(a) : a 9Ai}, then
h~(B) - {h(x) " x
9B }
9{f[(a') " a'
9A'i}.
By virtue of the corresponding definition, we have S ( A ) = {x~: I = (I1,/2) 9S}. Since B C S ( A ) and B 9{f/(a) : a 9Ai}, there exists an element a 9Ai such that
B = fi(a) = { x i : Let B' - { h ( x f ) Therefore
9
xI
a 9
9B}. If gi(a)
hi(B)* - {Y9(I)
(~-CIj, I = (11,/2)
~aVj -
a'
9Ai,' then a'-# 9~(Ii) ,,f----~,a ' Vj' e '
9a'-Egi(Ii) ~
g(I) - ( g l ( i l ) , g 2 ( I 2 ) )
9S}. -# 9y ( I j ) .
a' Vj' O ' ~ g j ( / j ) } ,
E S ' - - { h ( x I ) " xI E B }
and so h~(B) e {f/(a') 9a' c A~}. The pair h* - (h~,h~) is a G.isomorphism since g and f are G.isomorphisms. Indeed, let B' e {f;(a') 9a' e A~} be any set. Then there exists an element a' c A~ such that B' - f[(a'). Hence, there is an element a E Ai such that gi(a) = a ~ since gi are bijections. Let us consider the set f~(a) = B e {f/(a) : a e Ai}. It is obvious t h a t h'~(B) = B' so that h~ are onto. Now we assume that B1, B2 e {fi(a) : a e Ai} and B1 5r B2. Since f~ are bijections, there exist elements hi, a2 c Ai such that a l r a2 and fi(al) = B1, fi(a2) - B2. Let a~ - gi(ai). Then B~ - f~(al) , B; - f~(a~) imply h*(B1) - B~, h~(B2) - B;, and we have h~(B1) r h~(B2) because gi and f~ are bijections. Thus h~ are also bijections. It is trivial to verify the fulfilment of the following conditions: (1) h~(C1 ni (72) -- h~(C1)n~ h~(C2) and h~(C1 Ui C2) - h~(C1)U~ h~(C2) for each pair (C1, C2) 9(Ti n co Tj) X (Ti n co Tj). (2) h~(C1 ni C2) - h~(C1) N{ h~(C2) and h~(C1 Ui C2) - h~(C1)U{ h~(C2) for each pair (C1, C2) 9(rj N cori) x (ri N corj). (3) S ( A ' ) \ h~(C) - h~(S(A) \ C) for each set C 9r~ N co rj. Therefore h* = (h~,h~) is a G.isomorphism and thus ( f , S ( A ) ) and (f', S ( A ' ) ) are equivalent. Together with this equivalence, the definitions of the topologies r~ and r~ immediately imply that the BS's ( S ( A ) , T I , r 2 ) and (S(A'), r~, r~) are d-homeomorphic. This result means t h a t the G.isomorphy of GBA's implies the d-homeomorphy of the corresponding Boolean BS's. Next assume that (X, 71,72) is any Boolean BS satisfying the conditions of Theorem 6.8.2. W i t h (X, 71,72) we associate the GFS
oT'(X)- {(')'1 n co~'2)U (~2 n co')'1), n,u, \, ~, =:~, []1, []2} and its corresponding GBA
r
{~'1 nco~'2, n l , U l , \ , ==xQ,X,~2 n co~'l, n2, u2,\,}.
The GFS 2 ( X ) is the reduced one since (X, 71,72) is p-zero-dimensional and p-Hausdorff. We want to show that the Stone BS (S(A), r~, r2), associated with
6.8. Bitopological Representation ...
305
the Stone family of all prime GI's of the GBA A(a~(X)), is d-homeomorphic to (X, ~1, ")/2).
It is clear that the GFS Jr(X) -- {(~1 (-I CO ")/2) U (~2 A CO ~I) , ['-I, U, \, ==x(, []1, []2}, considered as the GBA
A(.~'(X))
--
{"71
I"'1
CO ")'2, ["ll, U1, \, 23, ==xQ, X , "/'2 f"l co ~1, A2, U2, \ },
is the reduced G.field representation of the same GBA A(.F(X)) in itself. Hence, by virtue of Theorem 6.7.4, with )c(X), considered as a G.field representation of A ( 5 ( X ) ) , one can associate the Stone family of prime GI's of A(Jz(X)). Therefore, applying the reasoning that precedes Theorem 6.8.1 with this G.field representation, we can associate a p-zero-dimensional and p-Hausdorff BS and, as is easy to see, this BS is d-homeomorphic to (X,'71,'72). But the BS (X,'71, ~72) is FHP-compact and, by Theorem 6.8.1, the corresponding Stone family S contains all prime GI's of A. Hence we conclude that (X, ~/1, ~/2) is d-homeomorphic to the Stone BS (S(A), T1, T2). The rest of the proof follows directly from Theorem 6.8.2. r-] D e f i n i t i o n 6.8.4. A G.lattice Z; = {L1,A1, Vl,@, m--X,e, L2, A2, V2} is said to be G.complete if its every subset has an ic-infimum and an i,~-supremum. Now we say that a GBA A = { A 1 , A 1 , V I , q P l , O , 4 , e , A2, A2, V2, qP2} is G.complete if the corresponding G.lattice {A1, A1, V1, 0, 4 , e, A2, A2, V2} is G.complete. L e m m a 6.8.5. Let f = (fl, f2)
: A
and A', where A is G.complete. Then
--+
G
A t be a G.isomorphism of GBA's A
fi ( i , - i n f B) - i , - i n f (fl ( U l ) U f2 (U2) ),
for every subset B c A1 U A2, where B rq A1 = B1 7k 23 7L B2 = B r~ A2, and, therefore, A' is also G.complete. Pro@ It suffices only to prove the equality fi(ic-inf B) = ic-inf(/1 (B1) U f2(B2)) since the proofs of both cases are similar. Since A = {A1,/~l,Vl,~91,l~,4 ,e, A2, A2, V2, qp2} is G.complete and f = (f~, f2) is a G.isomorphism, it is clear that A~ = {A~, A,, V,, ~&, (3, <_, e} are complete BA's and the fact that
f~. A~ - {A~, A,, v,, e,, e, _<, ~} -~ A'~ - { < , A,,l v,,l e,,l e , l ~I , (31} i
are isomorphisms implies that A{ -
i i I I I ~ i {A{, A~, V~, ~,, (3, , e'} are also complete i
BA's. Therefore L(infB~) = infL(B~), and it remains to show that (1)-(3) of Definition 6.1.6 hold for fi(i~-inf Bi) when they hold for ia-inf Bi.
306
VI. Generalized Boolean Algebra and Related Problems
(1) Since A is G.complete, for a subset B C A1U A2, there exists an ic;-inf B. Let b - f~(i~;-infB). Then by (1) of Definition 6.1.6, we have i~-infB <_ inf B~ and ic-inf B 4 inf Bj. Therefore b <_' fi(inf Bi) - inf fi(Bi) and b4'fj(inf By) - inf fj(Bj) i so that (1) holds. (2) Let x E A{, z _<' inffi(Bi) and x4'a for each a E fj(Bj). Since f i (fl, f2) is a G.isomorphism, that is, fl and f2 are isomorphisms, we conclude that f - 1 _ ( f l 1, f~-l) is also a G.isomorphism and, therefore,
f/-l(x) ~ f[-l(inf fi(Bi)) -- infBi, f/-l(x) ~ fj-l(a) i
for each element fj-l(a) c Bj. By (2) of Definition 6.1.6, f / - l ( x ) ~_ ic-infB and i thus x _<' fi(i~-inf B). i (3) Let x c Aj,r Xr 4 a for each a c fi (Bi) and x _<' inf fj (Bj). Then we easily J obtain fj-l(x) _< f 7 1 ( i n f f j ( B j ) ) - i n f B j , fj-l(x) 4 f ( l ( a ) J for each f ( l ( a ) c Bi. Hence, by (3) of Definition 6.1.6, fj-l(x) 4 iG-inf B and thus x4' fi(ia-inf B). Now, Definition 6.1.6 implies that fi(ic-inf B) -iG-inf(fl(B1 ) U f2(B2)). It is clear that if B c_ Ai, then, we have fi(i~;-infB) - i~-inff(B) and fi(ic-su p B) - / c - s u p f(B). [:] Our next theorem converts the property of a GBA .4 being G.complete into the property of the Stone BS (S(N), rl, r2) being p-exremally disconnected. Theorem 6.8.6. Under the hypotheses of Theorem 6.8.2, { A 1 , A1, V1, ~1, 1~, 4 , e, A2, A2, V2, ~2} is G.complete if and only
a GBA
A
-
if (S(A), rl, r2)
is p-extremally disconnected. Proof. First, we assume that A is G.complete. Then, by Theorems 6.8.2 and 6.8.3, the GBA r
{T1 n COT2, N I , U I , \ , { ~ , ==~,S(ce),T2 n COTI,N2, U 2 , \ } ,
which corresponds to the GFS .~'(S(A)) -
{(TI NCOT2) U (T2 NCOTI),N,U,\ , = = x ( , [ ] l , [ ] 2 } ,
is G.isomorphic to A. Hence, by virtue of Lemma 6.8.5, for every family of /-open and j-closed subsets of (S(A),rl, r2), there exists a smallest /-open and j-closed set containing all of them. Let U c r~ \ {~} be an arbitrary set. Since p-ind S(A) - 0, for each point x E U, there exists an/-open and j-closed neighborhood U(x) such that U(x) c U so that U - U U(x). Therefore for the
xEU
families of/-open and j-closed sets {U(x) 9 x EU}, there exists a smallest/-open and j-closed set U* such that U c_ U*. Thus rj cl U - r~ int rj el U - U* and, consequently, (S(A), rl, r2) is p-extremally disconnected.
6.8. Bitopological R e p r e s e n t a t i o n . . .
307
Conversely, let (S(A),T1,T2) be a p - e x t r e m a l l y disconnected BS and B c A 1 0 A2 be any subset. Let us prove t h a t there exist i ( j - i n f B and i c - s u p B . W i t h o u t loss of generality, we consider the case of i~-sup B. We have
B A A1 = B1 ~ ;g ~ B fB A2 = B2. Clearly, f l ( B 1 ) U f2(B2) C S ( A ) since f = ( f l , f 2 )
: A --~ A ( ) c ( S ( A ) ) ) is G
G . i s o m o r p h i s m and u
-
9
a
u
a
9
where {fi(a)
9 a ~Bi} c ~-i ~ c o T j
and { f j ( b ) "
b ~ By} c rj ~ c o T i .
It is obvious t h a t
U fi(a)-UEz-i, U fj(b)-VETj
aEBi
bEBj
and
Tj cl U = T~ int Tj cl U c T~ N COTj , T i c l V = v j i n t T ~ c l V c T j A c o T i since ( S ( A ) , T1, T2) is p - e x t r e m a l l y disconnected. Therefore, there exist elements a c Ai and b c Aj such t h a t f~(a) = Tj cl U, fj(b) = ~-~cl V and 7-i cl V Ui Tj cl U = [Ti cl V]~ Ui Tj cl U = 7-i int Tj cl Ti cl V Ui Tj cl U = 7i int Tj cl (~-~int Tj cl 7-i cl V U Tj int 7-i cl Tj cl U) = ~-i int (Tj cl 7-i int Tj cl 7-i cl V U Tj cl Tj int ~-i cl Tj cl U) c ~-i A co Tj since Tj cl ~-i int Tj cl Ti cl V U Tj cl Tj int 7-i cl Tj cl U C co Tj and ( S ( A ) , T1, T2) is p-extremally disconnected. Therefore T~ cl V U~ rj cl U c ~'i ~ co Tj is the smallest set containing fi (Bi) U f j (Bj) and so Ti cl V U Tj cl U = / ( ; - s u p ( f i ( B i ) U f j ( B j ) ) . By L e m m a 6.8.5, there exists i(.-sup B - f / - l ( i a - s u p ( f i ( B i ) U f j ( B j ) ) ) since f - 1 = (f~-l, f~-l) is also a G . i s o m o r p h i s m and hence ,4 is G.complete.
D
T h e o r e m 6 . 8 . 7 . For any G.component E of a G.complete G B A A = {A1, A 1 , ,e, A2, A2, V2, p2}, we have ai(L) = E = hi(R), where a = 1 , - s u p E and b = 2,:-sup E, that is, a = b V1 (~ and b = a V2 0 .
VI,~I,(:~),~
Pro@ Let E = E1 U E2 C A1 U A2 be a G . c o m p o n e n t , t h a t is, E = C d , (Cd(;E). Clearly, there exist a = 1 , - s u p E and b = 2(~-sup E. Then, on the one hand, by (5) of T h e o r e m 6.2.21, we have A z = C d , ( C d ( ; E ) and on the other hand, by (1) and (2) of T h e o r e m 6.2.22, we have aI(L) = A z = Cd~ (Cd(:E) = E = hi(R). T h e rest is obvious.
If]
308
VI. Generalized Boolean Algebra and Related Problems
R e m a r k 6.8.8. Let .4 = {A1, A1, Vl, (~91, t~, ~--~,e, A2, A2, V2, ~2} be a G.complete GBA, B = B~ U B2 C A1 U A2 and ~ ( g ) -- ~ I ( B 1 ) U ~ 2 ( B 2 ) ,
~)(B) = ~ ) l ( g l ) U @2(B2),
where i~,-infB-(
Aj x) Ai( Ai y) and / a - s u p B - ( x6 B 3
y 6 B~
Vj x) Vi( Ai y). xE B j
y ~ B.~
Then it is not difficult to see that
~( a~ x ) - v3 ~(x), ~( v~ x ) - a~ ~(x), x 6 B,,
x ~ B.i
~( /~ x ) x 6 B~
x ~ B,
v~ V~(x), r
x 6 B~
x 6 B~
v~ x ) -
x 6 B,i
/~ V~(x)
x 6 B.,
and, therefore, ic-infB-(
Aj x) Ai ( Ai y ) - x6B 3
= ~j((
y6Bi
Vi pj(x)) Vj ( Vj Pi(Y))) - ~ j ( j a - s u p ~ ( B ) )
x 6 B:i
y6 B i
so t h a t / a - s u p B = pj(jG-inf ~(B)). Similarly,
i~;-inf~ - ( =r
A~ x)A~ ( A~ ~ ) -
xEBj
Vj ~ j ( x ) ) V i ( Vi r xCBj
yCBi
~i(ie~-sup~(B))
y6Bi
so t h a t / a - s u p B = ~i(iG-inf ~(B)). The proof of Corollary 6.8.12 below will be facilitated by Remark 6.8.8. D e f i n i t i o n 6.8.9. A GBA A = {A1, A1, V1, Pl, (9, 4 , e, A2, A2, V2, ~2} is said to satisfy the countable G.chain condition if every G.disjoint system of non-zero elements of A1 U A2 is countable. T h e o r e m 6.8.10. A GBA .4 = {A1,A1,V1,q~)1,0,4,e, A2, A2, V2,~2} satisfies the countable G. chain condition if and only if every set E = E1 U E2 c A1 U A2 has a countable subset D such that u a (D) = % (E).
Proof. First, let B = B1 U B2 C A1 U A2 be a G.disjoint set (system) of non-zero elements of A1 U A2, D c B, IDI <_ b~0 and u c (D) = u G(B). If, for example, there is an element x c (B \ D) N A~, then d A~ x = (9 ~ d 4 ~ ( x ) for each d E D and hence pi(x) c % ( D ) = u~ (B), which is impossible since x 4 pi(x) is false. Thus B = D. To prove the converse, let the countable G.chain condition be satisfied and E = E1U E2 c A1U A2 be an arbitrary subset. If I = (I1, I2) is the GI generated by E, then u o ( E ) = u , ( I 1 U h ) . Indeed, always u~,(I1 U h ) c % ( E ) and if, for example, x c u ~ ( E ) N Ai, then y 4 x for each y c E. Let z c Ii be an arbitrary element. Then, according to Theorem 6.2.5, there are sequences al, a 2 , . . . , an C Ei and bl, b2,..., b,~ c Ej
6.8. B i t o p o l o g i c a l R e p r e s e n t a t i o n . . .
309
such t h a t z < al Vi a2 V~... V~ a~ and z ~ bl Vj b2
Vj ...
b,~.
Vj
Clearly, ak <_ x for each k - 1, n and bz ~ x for each 1 - 1, m so t h a t alvla2v1.-.vla~<_x,
bl Vj b2 Vj . . . Vj b,~ 4 x
i
and, therefore, z <_ x. But z c Ii is an a r b i t r a r y element and hence x c uc;(IltJI2), i
t h a t is, uc; (E) - uc; ((11 CJ/2). Furthermore, suppose t h a t 13 - { B ~ - B ~ U B ~ } ~ c T is a family of all G.disjoint sets consisting of non-zero elements such t h a t Ba C 11 [J/2 for each a E T. It is evident t h a t the family B is partially ordered by inclusion, t h a t is, Ba < BZ ~ B~ c B~ and if/C c 13 is linearly ordered, then ]C is b o u n d e d from above by the set B [.J Ba. It is not difficult to see t h a t B is a G.disjoint set B~6/E
of non-zero elements. Therefore, by Zorn's lemma, 13 contains a m a x i m a l element, say B0 - B ~ U B ~ t h a t is, a maximal G.disjoint set of non-zero elements. Since B1~ ~ C I1 U / 2 , we have u c ; ( I i U I 2 ) c uc;(Bo). On the other hand, let, for example, z C u ~ ; ( B o ) A A j , t h a t is, x ~ z ~ xAi~j(z) -- 0 for each x c B ~ ~ If z-Eu(;(I1 U I2), then, for example, there is an element y c I~ such t h a t y ~ z is false ~ ao - y A ~ j ( z ) ~ @. T h e n x A i a o -- @ since ao < ~ j ( z ) , where i
x C B ~ U B ~ is an a r b i t r a r y element.
Moreover, ao < y implies ao c I~ since i
I - (I1,I2) is a GI. Hence, we have found the element ao c Ii \ {@} such t h a t x Ai ao -- @ for each x c B1~ U B ~ This contradicts the m a x i m a l i t y of Bo and thus uc;(I1 U 12) - uc;(Bo), where Bo < Ro since the countable G.chain condition is satisfied. Clearly, E U Bo c I1 U/2. But we would like to find a set D c E such t h a t ]D[ < Ro and u~ (D) - u , (E). If x c Bo N A~ is an a r b i t r a r y element, t h a t is, if x ~ I~, then there are elements al,a2,...,a~
~ B~
bl,b2,...,bm
~ B~
such t h a t x < al Vi a2 V i - . . V~ a~ and x ~ bl Vj b2
Vj
bin.
".. Vj
i
Suppose t h a t D x --
{{al,a2,...
, an; bl, b2, . . .
,bin}
o
x <_ al Vi a2 Vi i
o
o
i
Vi an
and x ~ bl Vj b2 Vj . . . Vj bm }. Then [Bo[ _< Ro implies [D] _< Ro, where D -
U
Dx and u c ( D ) - uc~(B0 ) - uG(/1
U
12) - u(;(E).
[-1
x E B oi U B 2o
D e f i n i t i o n 6 . 8 . 1 1 . A G B A A = {A1,A1,VI,(~I,(~,4 ,e, A2, A2, V2,(P2} is said to be a G.Boolean a-algebra if its every countable subset has an ic-infimum (and, therefore, of course, and i , - s u p r e m u m ) .
310
VI. Generalized Boolean Algebra and Related Problems
It is clear that the notion of a G.Boolean a-algebra is an intermediate concept between that of a GBA and a G.complete GBA. C o r o l l a r y 6.8.12. A G.Boolean a-algebra .4 = {A1, A1, V1, ~ 1 , (~, 4 , e, A2, A2, V2, ~2} which satisfies the countable G.chain condition, is G.complete.
Pro@ Indeed, let B = B1 U B2 c A1 U A2 be any subset. Then according to Theorem 6.8.10, there exists a subset E = E1 U E2 c B such t h a t [E I < R0 and u(:(E) = u , ( B ) . Thus i(~-supE = / , - s u p B and by Remark 6.8.8, i(~-infE =
i~-inf B.
[-1 6.9. G e n e r a l i z e d B o o l e a n R i n g s
The notion of a G.Boolean ring is a special G.ring version of the notion of a GBA. D e f i n i t i o n 6.9.1. A G.ring is a non-empty set R - R1UR2 together with four binary operations | @i : (Rk x Ri) -~ Ri called G.multiplication and G.addition, respectively, which satisfy the conditions below: (1) There exists a unique element O E R1 N R2, called the zero element, such that x @~ (9 = (9 @i x = x for each x c Ri. (2) To each x E R~ there corresponds a unique element c~(x) c Rj such that x |
c~(x) = e .
(3) G.addition is G.commutative: xO~y=yOix
if (x,y) E R i x R i
(x@iy)-(y@jx)@iO
and
if (x,y) E Rj x Ri.
(4) G.addition is G.associative:
(x |
y) |
z = x e~ (y |
~)
if x, y, z E Ri or x E Rj, y, z E Ri and
(x e~ y)|
~ = x e~ (y e~ ~)
if x, y c Rj~ Z C R i o r x, z C R i , y E Rj. (5) G.multiplication is G.associative:
(x |
y) |
~- x|
(y |
z)
if x, y, z E Ri or x E Rj, y, z E Ri and
(~ |
y)|
z = x |
(y |
z)
if x, y E Rj, z E Ri or x, z E Ri, y E Rj. (6) G.multiplication is G.distributive with respect to G.addition:
x |
(y |
z) = (~ |
y) |
(x |
z)
y) |
(~ |
~)
(y |
z)
if x, y, z E R~ or x E Rj, y, z E Ri,
x | if
x, y
c Rj,
(y e~ z) = (x |
Z C R i o r x, z C R i , y
(x |
y) |
~ = (x |
c Rj, ~) |
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6.9. G e n e r a l i z e d B o o l e a n Rings
317
(2) hi o Oi - ED~ o (hk, hi) so that hi(x Qi Y) - hk(x) O'i hi(y) for each pair (x, y) e Rk x R~. (3) c', o hi - hj o ci so that c{(hi(x)) - hj(ci(x)) for each x 9Ri. A G.homomorphism h = (hi,h2) of GBR's is a G.isomorphism if hi and h2 are bijections. It is not difficult to ascertain that the following statement is true. T h e o r e m 6.9.7. Let A be a GBA, 7g(A) be the GBR, constructed from A by the rules stated in Theorem 6.9.4, and let A(Tg(A)) be the GBA constructed from 7g(A) by the rules stated in the same theorem. Then A is G.isomorphic to A(Tg(A)) in the sense of Definition 6.4.6. Similarly, ifTg is a GBR, A(Tg) is the GBA, constructed f f o m 7g, and 7~(A(7~)) is the GBR, constructed from A(Tg), then 7g is G.isomorphic to 7g(A(Tg)) in the sense of Definition 6.9.6.