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Chapter 5 Filters, Ideals and ϑ-Congruences
247
CHAPT E R 5
FILTERS, IDEALS AND 8-CONGRUENCES
T he aim of this chapter is t o construct a theory of filters, ideals and congruences in LM-algebras generalizing the main results of t h e correspon-
ding theory for distributive lattices (6.
Ch. 1, 93).
T h e starting point is
a convenient definition o f th e concepts of filter, ideal and congruence in LM-algebras, obtained by adding appropriate extra conditions to the corresponding concepts of lattice theory and universal algebra, respectively (cf.
Ch. 1). T he last section answers a problem raised by Cignoli i n connection wit h a property o f prime filters in Post algebras.
$1.d-filters, d-ideals and 8-congruences
A 9-filter (&ideal) of an LM-algebra is quite naturally defined as a filter (an ideal) compatible w i th the endomorphisms cpi. It turns o u t t h a t &filters coincide wit h Stone filters and w i th deductive systems. Since the quotient
L / p of an LM-algebra L by a congruence p (in the sense of universal algebra) need not satisfy th e determination principle, we introduce t h e stronger concept of 9-congruence. T h e “metatheorem” of this section is t h a t 6- filters (&ideals),
d-congruences and quotient algebras behave as one would
expect. See also Hatvany [1984].
1.1. Definition. Let L E LMd. A d-filter of L is a filter F of L such t h a t A d-ideal of L is an ideal H of
x E F jcpox L such th a t x E H + cp1z E H .
E
F.
In the sequel we only deal w i th &filters b u t all the results can be transferred by duality to 1’1-ideals.
Filters, ideals and 6-congruences
248
1.2. Remark. For every &filter
F
The concept of &filter was introduced by Moisil [1940], [1963c] for nalgebras and later [1968] for d-algebras, under the name of strong filter (because there exist filters that are not &filters, e.g.
F
#
(1) of L d : if x E F - (1) then pox = 0
proper d-filter of
Liq is (1)). However,
#
every proper filter
F ; similarly the unique
it turns out t h a t in the case of
LM-algebras 6-filters coincide with Stone filters introduced by A. Monteiro
[1954] (6. A. Monteiro [1974]) and studied by Cignoli [1966a], [1966b] and Cherciu [1971].
1.3. Definition.
A Stone filter of a bounded (distributive) lattice L is a filter F of L such that for every x E F there is y E F n C ( L ) such t h a t y _< z. We set
F* = F n C ( L ) .
(1.1)
1.4. Proposition.
The following conditions are equivalent for a filter F of an algebra L E LM6: (i) F is a 6-filter; (ii) F is
a
Stone filter;
(iii) F is generated by F*; (iv) Proof.
F = cp;l(F*).
+ (ii): For x E F take y = pas. + (iii): Obvious. + (iv): Because z E F * pox E F H 'pox E F*
(i) (ii) (iii)
249
&filters, 19-ideals and 19-congruences (iv) =+ (i). Because x E
F =+ 'pox E F' G F.
Comment.
It is easily seen that F* is the unique filter of C(L) satisfying (iv).
1.5. Definition. Let L E LM9. For every subset X of L we denote by [ X ) , the &filter generated by X , i.e. t h e least 19-filter including X . Also, we denote by Fl(L) and F119(L) the lattice of all filters of L and t h e lattice of all 19-filters of L, respectively.
1.6. Remark. Let L E LM19. Then [0)4 = (1) while if (1.2)
[XIS
= {Y E L I 'po(zi A
I
0# X
L then
... A xn> I
y;sl,...,2n~~;n~mr-{0}};
C C ( L ) then [ X ) , = [ X ) while if X is a filter of L then [X>S= {Y E L I 'pox I y, 2 E X } .
in particulx if X
1.7. Theorem.
For every L E LM19 the m a p s * : F119(L) + Fl(C(L)) defined by (1.1) and 'pi' : Fl(C(L)) -+ F119(L) are lattice isomorphisms anverse t o each other. Proof. Clearly the maps * and 'pol are isotone and their targets are those indicated above. The composites
'pi'
o * and * o 'pi' are the identity
mappings by Proposition 1.4 and
g,'(P) n C(L) = P which follows from x E
( V P E FlC(L)) ,
C(L)n 'p;'(P) -s x = 'pox E P -s 5 E P .
The congruences in the classes LMQ and LMNQ are introduced according t o the general Definition 1.5.11. However, compatibility with 9; and N follows from the other conditions:
Filters, ideals and 8-congruences
250
1.8. Proposition (Boicescu [1984]). T h e following conditions are equivalent f o r a n equivalence relation p of a n algebra L E L M 8 :
(i) p is a congruence; (ii) p is compatible with A, V and
'p;
(Vi E I ) .
Proof.
(i) j (ii): Trivial. (ii) + (i): We suppose x p y and prove CpixpCpiy. But ' p i x p p i y hence 1p('piy V p i x ) and similarly 1p ('pix V Cpiy). This implies in turn
and similarly piypCp;x A Cpiy, therefore p i x p p i y
(Vi E I ) .
0
The construction of quotient algebra requires a stronger concept (see Remark 1.16). 1.9. Definition (Boicescu [1984], lorgulescu [1984c]).
A 29-congruence of an algebra L E L M 8 or L E LMN29 is a congruence of L E DO1 such that x p y ('pix p (piy (Vi E I ) ) . 1.10. Proposition.
T h e following conditions are equivalent f o r a binary relation p of a n algebra L E LM29;
(i) p is a 29-congruence; (ii) there is a congruence p' of L such that z p y Proof.
(i) + (ii): Take p' = p.
@
(Pix p ' v i y (Vi E I ) ) .
d-filters, 19-ideals and &congruences
251
(ii) + (i): Clearly p is a congruence and if 'pjxpcpjy (Vj) then 0
cpicpjx p'(pj(pjy ( V i , j ) , therefore (pjx p'(pjy (Vj), i.e. z py.
1.11. Definition. Let L E LM29 or
L E LMN29. For every congruence p of L we denote by
p the d-congruence
generated by p , i.e. the least d-congruence including p.
O ( L ) and Oo(L) the lattice of all congruences of L and the lattice of all &congruences of L , respectively. Also, we denote by
1.12. Remark. Let L E LMr9 or L E LMNd and p E z P Y @ (piz ppiy
(1.3)
O ( L ) . Then
(vi E 1))
*
To establish t h e relationship between d-filters and d-congruences we intraduce the maps kerl;, modl; and modl;, or simply ker, mod and mod: (1.4)
ker : Oo(L) --t F119(L) ,
(1.5)
mod : F119(L) --t O ( L ) ,
(1.6)
mod
: Fld(L) + Oo(L)
defined by (1.7) (1.8) (1.9)
ker p = {z E L I z p l }
(Vp E Oo(L)) ,
z m o d F z ' @ 3u E F z A u = z' A u
- mod =
o mod
(VF E Fld(L))
,
(cf. Definition 1.11) ,
respectively.
1.13. Theorem (Boicescu [1984],lorgulescu [1984c]). Let L E LM19 OT L E LMN.9. The maps mod and ker satisfy mod = ker-' and establish an isomorphism F129(L) S Oo(L). Proof. It is easy t o check that ker p E Fld(L) even for p E O(L); e.g. if z E ker p and z
5 y then y = z V y p 1V y = 1i.e.
y E ker p. Let us prove
252
Filters, ideals and 19-congruences
that m o d F E O ( L ) for F E F119(L). Take x m o d F x ' a n d y m o d F y ' i.e. x Au = x'Au a n d y A v = y ' A v for u,v E F. Then (PiXAviu = (PiYAviu where (piu E F, hence 'pix mod F'piy, while from x A u A v = x' A u A v and y A u A v = y' A u A v where u A v E F, we deduce immediately that ~ A y p ~ ' A y ' a nxVypx'Vy'. d If L E LMN6 then N x V N u = Nx'VNu. But from
we get N u A 'pou = 0 hence
NXA YOU = ( N x V N U )A C ~ O U = ( N d V N U ) A
YOU
=
where pou E F, showing that Nxmod F Nx'. Clearly the maps ker, mod and are increasing. To prove ker o m o d = 1 F l q ~ take ) F E Fld(L). Then
-
-
ker(modF) = {x E L zmodFI} = = { X E L vix mod F 1 (Vi E
I)}
=
= {XEL
by Remark 1.2. To prove mod o ker = 1e0(,qnote first that you 5 u and ker p n C(L)= ker(p n (C(L))')and recall Proposition 1.3.26. Then for every p E Oo(L)
zmod
o ker p y H pixmod ker pviy
(Vi E I)
19-filters, 19-ideals and d-congruences
253
1.14. Proposition (Boicescu [1984]). Let L E LMn or L E Mn. T h e n Oo(L)= O ( L ) and the maps mod and ker satisfy mod = ker-' and establish a n isomorphism Fln(L) O ( L ) . Proof. Let p E
-
O ( L ) . If x mod o ker p y then (i = 1,...,n - 1)
cpixmod o ker pcp;y
w (3u; u ; p l & cpix A ui = (piy A ui) therefore taking u = u1 A
(i = 1,...,n - 1) hence
... A u,-'
(i = 1, ...,n - 1)
we obtain u p 1 and cpix A u = (piy A u
(i = 1,...,n - 1) , therefore x A u = y A u, showing that
mod o ker p therefore
(1.10)
3 mod
-
o ker p y. Thus mod o ker p
E
-
mod o ker p = m o d o ker p E Oo(L) .
Taking into account Theorem 1.13 and (1.10) we have
-
xpy+xpy*xmod (j
o kerpy
cpixmod o ker pcp;y
w
(i = 1, ...,n - 1) e
* (3% u; p 1 & pix A u; = cp;y A u;)
(2
= 1,...,n - 1)
+
+- (3% (o;u~p 1 & c p ; ~A ~piui= (P;Y A 9;";)
(i = 1, ...,n - 1)
+ (i = 1,...,n - 1)
+
=$
(pix mod o ker py;y
=$
xmodokerpyw3uupl&xAu=yAu
+- 3~ x = x A I p s A U = Y
A U ~ YA 1 = y
proving that p = mod o ker p, therefore p
O(L)= Oo(L), hence by Theorem 1.13.
-
+. X
+ ~
Y
E Oo(L)by (1.10). This proves
is the identity mapping and the proof is complete 0
254
Filters, ideals and d-congruences
1.15. Corollary.
I f B is a Boolean algebra t h e n Oo(B)= O ( B ) . F12(B) = FI(B) and the m a p s mod and ker satisfy mod = ker-' and establish a n i s o m o r p h i s m Fl(B) O ( B ) . 1.16. Remark. Definition 1.9 ensures that if p is a 29-congruence of an algebra L E LM6 ( L E LMN29) then the determination principle holds in L / p , therefore L / p E LM19 ( L / p E LMNB). In view of Theorem 1.13 every &congruence p is of t he form mod F for some &filter F (= ker p). The quotient 29algebra L / m o d F will be denoted simply by L / F . In particular if L E LMn or L E Mn then every n- congruence is of t h e form m o d F and L / F will stand for L/mod F .
1.17. Remarks. a) In a Moisil n-algebra Proposition 1.8 remains true: If L E Mn we still
have t o prove that x p y implies N x p N y . This follows from
Therefore a l l types o f congruences reduce t o lattice congruences compatible with 'pi.
b) However, this does not hold in general, as shown e.g. by the following example due t o lorgulescu 11984~1.Suppse I is an infinite regular set (6. Definition 4.6.7) and take L = (?(I))['. Let Fo C C ( L ) be the filter of those functions f E C ( L )for which the constant value f ( i ) is a cofinite set (i.e., I - f ( i ) is finite). We denote by F the &filter generated by Fo and prove that mod F is not a &congruence. Let g, h E L be defined by g ( i ) = [O,i-]C I for i # 0, g(0) = 0 and h ( i ) = [O,i]G I,for Vi E I . For each i E I, define f; E C(L)by f i ( j ) = I - {i} V j E I. Then f i E FO C F and ('pig
fi)(j)
= g ( i ) n f i ( j ) = [O, i-]
n ( I - {i})
= [o, i-]
&filters, 19-ideals and d-congruences
255
and similarly ((pjhA f i ) ( j ) = [0,i-1, therefore (pig A fi = (pihA fi, proving that YigmodFcpih for every i E I . It remains to show that ( g m o d F h )
does not hold, i.e. that g A
f # hAf
for every
f E F.
f E F fo 5 f.
But
implies, via Remark 1.6, the existence of fo E Fo such that
This implies further the existence of a finite subset loc I such that 8 # I - I0 = fo(i) 5 f(i) Vi E I. Take i E I - Io; then g A f # h A f follows from
1.18. Remark.
It follows from Remark 1.17 b) and Proposition 1.8 that for certain sets 1 the corresponding classes LMd and LMNd are not equational (because the homomorphic image L / p of L is outside t h e class whenever the congruence p is not a &congruence).
1.19. Proposition. Let L E LMd or L E LMNd. If F, F’ E Fld(L) and F 5 F‘ then = {i I x E F’} is a 29- filter of L / F . If, moreover, F’ is proper (prime) is so. then
F?
Proof. The first statement is obvious. If 0 E
u; E F (Vi E I ) , hence 0 =
(pis A
then 0 = ( p i x A ui, s E F’,
(p;ui, which implies (piu; 2 p i x E F‘
F‘ and since (p;ui E F C_ F’ it follows t h a t 0 E F’. If 3 V E F? then pi(^ V y) A ui = (pi2 A U i , z E F‘, ui E F (Vi E I ) , hence ( p i x A U i ) V ( v i y A U i ) E F’, implying e.g. (pox A uo E F’, in which case x E F‘ because x 2 (pox A ug, therefore i E F? otherwise 6 E F?. O
therefore piui E
1.20. Remark (lorgulescu [1984c]). The concept of d-filter and th e concept of m-filter in Remark 1.3.31 are independent of each other. Thus e.g. a) if
I
23 then the filter I of Ld = J = (0) + I is neither a &filter nor an m-filter. Further b) let E be a set of cardinality m and the filter F of P(B)consisting of all cofinite sets.
Filters, ideals and &congruences
256 Then E - {z} E
F
for every z E
E and
n
(E- {z})
xEE
= 8 !$ F . Take
L = B[q and let F' be the image o f F by the isomorphism B E C ( L ) . Then [F')s= {f E L 139 E F', g 5 f} by Remark 1.6 and this &filter is not an m-filter. Now c) take an m-algebra L and a E L . Then [ a ) is an m-filter; besides, [a) is a &filter or not according as a E C ( L )or not. A filter which is both a &filter and an m-filter will be called an (m, O)-filter. The reader is urged t o transcribe the results o f this section to the case of the (m, d)-filters of an m-algebra. In particular the restriction of the functions in Theorem
1.7 yields an isomorphism between the complete lattice o f all (m, d)-filters of t he m-algebra
L and the m-filters of the Boolean m-algebra C(L).
1.21. Remark (lorgulescu [1984c]). Let us call m-congruence o f an LM-algebra
L , any congruence relation p on
L satisfying the stronger conditions (i) for every
A
h€H
L , c a r d H 5 m, if a h p b h (Vh E provided both meets exist, and
( a h ) h € ~ (, b h ) h E ~C
ahp
A
bh
H ) then
h€H
(ii) the dual condition. Suppose L is an m-algebra (cf. Definition 4.6.1) and let F be a &filter of
L. Then using Proposition 1.2.27 and F = i it is easily seen that m o d F is an m-congruence if and only if F is an (m,d)-filter. Let us call (m,d)congruence every 8-congruence which is also an m-congruence. Thus m o d F is an (m, 29)-congruence for every (m, 9)-filter
F
and as a matter of fact the
isomorphism constructed in Theorem 1.13 can be restricted to an isomorphism between the complete lattice of all (m, 29)-filters and the complete lattice of all (m,d)-congruences.
The reader is urged to transcribe other
results of this section t o the case of (m, 8)-congruences. Now we turn to further consequences of the previous basic results. Proposition 1.14 has a kind of converse for 3-valued Moisil algebras. Let
L
be a double pseudocomplemented lattice; by a ++-closed filter is meant
257
8-filters, 8-ideals and 29-congruences a filter
F of L such that x E F
+ x++
++-closed filter for every congruence p of
E F. It is obvious that ker p is a
L.
1.22. Theorem (Varlet [1972]).
Let L be a double pseudocomplemented lattice. T h e m a p ker is a bijection between O(L)and the set of ++-closed filters of L if and only i f L E M3. Sketch of proof. Sufficiency. If L E M 3 then i t s congruences coincide with
the congruences of the underlying double pseudocomplemented lattice, while the 3-filters are exactly the ++-closed filters. Necessity. In view of Theorem 4.4.4 it suffices t o show that
-
L is a dou-
ble Stone algebra verifying the determination principle. The latter property follows from the fact that the congruence
defined by (z
N
y
+ x* = y*
and z+ = y+) satisfies ker N= (1) = ker
1~ hence -= 1 ~ Further . one proves that if x+ A x++ > 0 then the ++-closed filter [zt)is distinct from ker p for every p E O ( L ) . This contradiction shows that zt A z++ = 0 and 0 one proves similarly that z* V x** = 1. 1.23. Proposition.
If L E LMn t h e n O(L)E O ( C ( L ) ) hence O ( L ) is a Heyting dgebra. Proof. Follows from Theorem 1.7, Proposition 1.14, Corollary 1.15 and e.g.
the dual of Example IX.3.5 in Balbes and Dwinger [1974].
0
The next two propositions were first proved by Beazer [1976] for 3-valued Moisil algebras, using the theory of double palgebras. 1.24. Proposition.
T h e following conditions are equivalent for L E LMn:
(i)
O ( L ) i s a Boolean algebra;
(ii)
C(L)i s finite;
(iii) L i s finite.
Filters, ideals and d-congruences
258 Proof.
(i) ($ O ( C ( L ) ) E B H (ii): By Proposition 1.23 and Berman [1973]. (ii) + (iii): Because L' = (C(L))["-'] is a finite Post algebra and PL : L --f L' is a monomorphism. (iii) + (ii): Trivial. 0 1.25.
ProDosit ion.
T h e following conditions are equivalent foT L E LMn:
(i) @ ( L )is a Stone algebra; (ii) C(L)i s a complete Boolean algebra. Proof. (i) H
O ( C ( L ) ) is a Stone algebra
H (ii) by Proposition 1.22 and
Beazer [1973].
0
1.26. Definition.
A class K: of 7-algebras is said to have the congruence extension property if for every A E K , every subalgebra B of A and every congruence p of B , there is a congruence p' of A such t h a t p'
n B2 = p.
1.27. Proposition (Boicescu (19841).
T h e class LMn has the congruence extension property. Proof. Let A E LMn and B , p as in Definition 1.26. The filter F generated by ker p in A is obviously a d-filter and we shall prove that m o d F n B 2 = p. If ( z , y ) E m o d F f l B2 then x , y E B and there is u E A such that
x A u = y A u and a 5 u for some a E ker p , therefore x A a = y A a, hence ( 2 , ~ )E mod o ker p = p by Proposition 1.14. Thus m o d F n B 2 E p, while 0 the converse is obvious from p = mod o ker p. 1.28. Remark.
It is known (see e.g. Pierce [1968], Lemma 3.1.8) t h a t if A is a 7-algebra and p,u E @ ( A )then p o u E @ ( A ) if and only if p o u = a o p, in which case p o u = p V u in the lattice @ ( A ) .The algebra A is said t o have
259
29-filters,29-ideals and 10-congruences
commuting congruences provided p o a = a o p for every p , a E @ ( A ) . 1.29. Theorem (Boicescu [1984]; cf. Beazer [1972] for M3).
Every L E LMn has commuting congruences. Proof. Let p l , p 2 E @(L).Then p h = modFh ( h = 1,2), where Fl and F .
L. Take (x,y ) E p1 o p2. Then there exist z E L and uh E Fh 1,2) such t h a t x A u1 = z A u1 and z A ug = y A u 2 . This implies
are n-filters of
(h =
x p l x A u 1 , yp2 yAu2 and x A u l A u 2 = y A u 1 A u 2 , therefore x A u l p 2 y A u l
x A uzp1 y A u2. Setting w = (x A u2) V ( y A u l ) and using t h e above
and
relations it follows that
i.e. x p ~ ~ a n d s i m i l a r l y y ptherefore(x,y) ~v, E p20p1. Thusplop,
and
p2
o p1
C p1
p20p1
o p2 by symmetry.
1.30. Remark. It is known that for every filter F of a Boolean algebra B and every x,y E B ,
(1.11)
x mod F y
* (x V jj) A ( 5 V y) E F
(x V jj) A (3 V y ) E F then x A u = x A y = y A u and conversely, if x A u = y A u for some u E F then using (2.18") we get because if u =
x V y V G = ( x A u ) V Y V G = ( y A u ) V j j V i i = 1, hence u (x
v
5 x
V y by (2.19") and similarly u
5
5 V y , therefore u
E F . This property was generalized by Varlet [1972] matter of fact it holds for LMn.
y) A (2 V y)
t o LM3 and as a
1.31. Proposition.
Let L E LMn (1.12)
OT
5
L E Mn. FOTevery F E Fln(L):
zm o d F y
*
n-1
A (pix V 9 ; ~A )
i=l
V piy) E
F
Filters, ideals and 19-congruences
260
and every congruence p i s obtained in this way: p = mod o ker p. Proof. Using Proposition 1.14 and Remark 1.12 we get
zmodFy @
* z modFy * (i = 1,...,n - 1) ,
cpizmodFcp;y
whence (1.12) follows by Remark 1.20. The equality p = mod o ker p was obtained in the proof of Proposition 1.14.
1.32. Definition. Let us say that: a) An element
z of a .r-algebra A has property R ("regular") provided the
map @ ( A )+ ?(A) defined by p b) An algebra A has property
H
[z],, is injective;
R whenever all of its elements have property
R; c) A class
K: of valgebras has property R if all of its algebras have property
R. Varlet [1972] proved t h a t a double pseudocomplemented lattice has pro-
R if and only if z* = y* & z+ = y + + z = y . In particular L M 3 has property R and this characterizes L M 3 within t h e class of double Stone
perty
algebras. One can prove more: 1.33. Proposition.
The class LMn has property R. Proof. Notice first that the element 1has property R, because if [l]= ,, [1lPz then ker
p1
= ker
p2
whence Proposition 1.30 implies
p1
= p2.
Then take z E L E LMn and p I , p 2 E O ( L ) such that [z],,
= [z],,.
1 (cf. Theorem 4.4.16) one sees Using the LMn-structures of [0, z]and [z, 1 that
pi
= P k fl [ O , Z ] ~
P k fl [z, 11'
(k = 1,2)
(k
= 1,2) are congruences on [O,z] and pk" =
are congruences on [z,11. Furthermore
261
&filters, 8-ideals and 8-congruences
and since x is the unit o f [0, x] it follows that pi = pk and similarly ply’ = p2”.
Now take (u, v) E p1. Then (u A z, v A X ) E pi = p; and ( u V x, v V z) E
p2”, therefore u = u V ( u A X ) pz u V (v A z) = ( u V
proving that ( u , v ) E p2. Thus
p1
v) A
(u V x) p2
C pz and similarly pz
0
pl.
1.34. Remark.
The concept of filter in a Boolean algebra has been given an equivalent definition, very appropriate t o the purposes of symbolic logic: a filter is the same as a deductive system. The latter term designates a subset
D
of the
Boolean algebra B such that
(i) 1E D and (ii) if z E
D
and z + y E
D
then y E
D.
B . Then 1 E F and from x E F and x -+ y =? vyi E F we obtain z ~ = yx A ( % V Y ) E F hence y E F . Conversely, suppose D is a deductive system. Then D # 0 because 1E D. If x E D and x 5 y then x ---f y = 1E D therefore y E D.
To prove the equivalence suppose first that
F is a filter
of
If x , y E D then from
x
-+
x A y = Z V ( x A y ) = f V y L. y
we obtain z + x A y E D by the previous property, therefore x A y E
D.
It turns out that the above properties can be extended t o LM-algebras. The point is a convenient generalization of the Boolean implication, invented by A. Monteiro in 1963 for 3-valued Moisil algebras and called weak implication; cf. Cignoli [1969a]. Then Cignoli [1969a], [1970] defined and studied the weak implication and the deductive systems of Moisil n- algebras. Thus the results presented below are due t o A. Monteiro and Cignoli.
Filters, ideals and d-congruences
262
-
1.35. Definition.
-
The weak implication on an LM-algebra is the operation
x
M
defined
y='PoxVy(Vx,yEL).
M
- -
1.36. Proposition.
The weak implication
has the following properties:
M
(1.13)
x
(1.14)
x
(1.15)
x
(1.16)
x
(1.17)
xVy
(1.18)
(x
(1.19)
1
(1.20)
z 5 y implies x -+
(1.21)
x 5 y iff (pix
(1.22)
2
(y
M
z)=l;
M
+ M
M
M
(y
M
y A z = (x
- -
M
Z = ( X
y)
M
Y)+
+ M
+ z ; M
z)A(y
+ M
2 = 2 ;
x=x;
--t
M
QoX=1.
Proof. Straightforward .
y =1 ;
(piy= 1
z);
M
M
M
--t
M
M
M
-
- M
(y + z ) = z A y
--t
M
; y) + (x
z)=(x
+
, Vi E I ;
z);
+ M
z);
by
263
6-filters, 6-ideals and 6-congruences
1.37. Definition. If X is a non empty subset of L then we say that x E L is a consequence of m
X if there is a finite subset { x l , ...,xm} of X such that A
j=1
xj + M
x = 1.
Let us denote by C s ( X ) th e set of all consequences of X ; we put also CS(0) = (1).
1.38. Proposition.
C s ( X ) = [ X ) , for every subset X & L E LMn. Proof. Cs(0) = (1) = [0)8. Then
m
U
~
A
O j=1
X j S Y
and the proof is completed by Remark 1.6.
1.39. Proposition. Let L E LMn. A subset S
0
L is a deductive s y s t e m if and only if it is
a 6-filter.
-
Proof. Let S be a deductive system. Then S m
x E Cs(S) then
A
j=1
xj
M
# 0
because 1 E S. If
x = 1 for some x l , ..., xm E S, therefore
Filters, ideals and &congruences
264
...
z1 + (zZ + M
M
-+ M
...) = 1 by (1.16),
-+ z)
(I,
M
hence after
c
m steps we obtain z E S. Thus Cs(S) C S and since S [S)e = Cs(S) by Proposition 1.38, it follows t h a t S = Cs(S) = [S)s is a &filter. Conversely, let S be a d-filter. Then 1 E S. If z E S and z then
(POX
-
M
yES
E S n C ( L ) = S' and
(Pox
(POY
M
-
= pox V(P0y = (Po(p0z v y ) = = (Po(z
M
E S' ;
y)
9 is obviously a filter of th e Boolean algebra C ( L ) ,hence a deductive system of C ( L ) by Remark 1.34, therefore 'pay E S c S, whence finally y E s. 0 but
1.40. Proposition (the Deduction Theorem).
Let L E LMn, X (1.23)
L
and z, y E
L. Then
y E C s ( X U {z}) H z + y E C s ( X ) . M
Proof. We use Proposition 1.38, Remark 1.6 and the obvious property you
wH
(POU
I (POW.
F
Let
show that F = { y E
= {y E L I z
--f
L 1 cp0(zl A ... A z,
E C s ( X ) } . The equivalences
5 'pay}, whence it is straightX U {z} F and F E F' for any F = [XU (5))s. The above proof i s
F is a $filter, &filter F' such that X F'. Thus also valid for X = 8 if we replace z1 A
forward t o show t h a t
y
M
5
A z)
... A 2,
by 1.
265
Prime filters $2. Prime filters
The aim of this section is the study of prime filters and prime d-filters in a Lukasiewicz-Moisil algebra. A special attention is paid to minimal and maximal prime filters. 2.1. Definition. Let L f LMd. For
L and G
F
(1.1)
F* = F n C ( L ) ,
(2.1)
G;= cp;’(G)
C ( L )we set
(Vi E I ) .
A proper (prime) d-filter is defined as a d-filter which is also a proper (prime) filter. We denote by PFld(L) the set of all prime 8-filters on L. By a masimald-filter is meant a maximal element in the family of proper 8-filters ordered by set inclusion. 2.2. Lemma (Cignoli [1969a]).
Let L E LMd, F a filter on L and G a filter on C(L).Then for every i ,j E I : (i)
G;is a filter on L ;
(ii) i (iii)
+ Gi
Gj;
(G;)*= G;
(iv) F* is a filter on C ( L ) and (F”)o
zF c (F*)l;
(v)
G is prime, or equivalently maximal, in C ( L )($ Gi is prime in L ;
(vi)
if F is prime in L then F* is prime, or equivalently maximal, in C(L).
Proof. Straightforward.
266
Filters, ideals and 29-congruences
2.3. Theorem (Cignoli [1969a], Georgescu and Vraciu [1969a], Georgescu
[197I d ]). Let L E LMd and F a 19-filter o n L. T h e n the following conditions are equivalent:
F is a m a z i m a l O-filter;
(i)
(ii) F* is a n ultrafilter o n C ( L ) ; (iii) F* is a p r i m e filter o n C ( L ) ; (iv) F is a prime 8-filter; (v)
either
(vi)
F is
'piz
E
F
OT t&a:
E F (Vi E I) (Vz E L ) ;
a m i n i m a l p r i m e filter.
Proof.
(i) H (ii): By Theorem 1.7. (ii) e (iii) e (v): By Proposition 1.3.18. (iii) e (iv): Because z V y E F e 'pox V 'poy E F*. (iv) (vi): Take a prime filter F' of L such t h a t F' c F . Then F'* c F* and F'* and F* are prime filters of C ( L ) by Lemma 2.2, hence they are maximal by Proposition 1.3.18, therefore F" = F*. Now it follows by Proposition 1.4 and Lemma 2.2 that F = (F*)o= (F'*), c F', therefore F = F'. (vi)
j
(iv): Trivial.
0
2.4. Remarks.
F is necessarily a &filter, because Lemma 2.2 implies in turn that (F*)o& F , F* is prime in C ( L ) , (F*)o is prime in L , therefore F = (F*)ois a &filter by Theorem 1.7.
a) Any minimal prime filter
b) Prime (or, equivalently, maximal) &filters coincide with minimal prime filters, by Theorem 2.3 and t h e above remark a). c) The map * establishes a bijection between PF16(L) and PFl(C(L)), by
Theorems 1.7 and 2.3.
Prime filters
267
Several characterizations of those lattices
L E DO1 in which maximal
Stone filters coincide with minimal prime filters were obtained by A. Monteiro [1954] and then by Cignoli [1971b], who started from a result of Cherciu
[1971]. 2.5. Proposition.
Suppose L E LMd, F as a 29-filter, 0 # S C L , F n S = 0 and x,y E S + x V y E S. Then there is a prime 19-filter P such that
PnS=Q)andFGP.
F* n (cp0S]c(~) = 0, otherwise x 5 cpos for some x E F* and s E S, which would imply cpos E F , then s E F , a contradiction. According t o Q and Theorem 1.3.10 there is a prime filter Q of C ( L ) such that F* 0 Q n (cpOS]c(~)= 0. Then P = Qo has the desired properties. Proof.
2.6. Remark. Proposition 2.5 yields corollaries similar t o Corollaries 1.3.11-1.3.14 of Theorem 1.3.10. Thus e.g. a) in a d-algebra every proper d-filter is included in a prime &filter and is
an intersection of prime &filters. Also,
b) given z,y E L with x
#
y, a necessary and sufficient condition for the existence of a prime d-filter F such that x E F and y $ F is cpox $ y.
2.7. Proposition.
The intersection of all maximal 9-filters of a n LM19-algebra L is (1). Proof. It follows from Theorem 2.3 that an element x belongs t o all maximal 6-filters
F of L if and only if cpOx belongs to all maximal filters F* of C(L),
and this is equivalent t o pox = 1 by Theorem 1.3.28, therefore t o x = 1 by (3.1.27) and (3.1.2). 2.8. Definition. Let us say t h a t a congruence (a d-congruence) is:
(i) proper if it is distinct from the universal congruence L2,and
0
268
Filters, ideals and 6-congruences
(ii) maximal if it is a maximal element in t h e family of all proper congruences (proper 2')-congruences)ordered by set-inclusion
C.
According t o a general definition of universal algebra, an algebra A i s called semisimple if the intersection of all its maximal congruences is the trivial
congruence A.A. 2.9. Corollarv.
The i n t e r s e c t i o n of a l l maximal 19-congruences of an LMb-algebra L i s
AL. Proof. From Proposition 2.7 and Theorem 1.13.
0
2.10. Corollary. LMn-algebras and Mn-algebras are semisimple. Proof. From Corollary 2.9 and Proposition 1.14.
0
2.11. Remark.
LM6 (LMN1S)-algebras are also semisimple, because maximal 6-congruences are identical with maximal congruences. 2.12. Proposition.
The following conditions are equivalent f o r a prime ideal I of an algebra
L E LM6: (i)
I
(ii)
In Ds(L) = 0;
i s a 2')-ideal;
(iii) for every x E L, e i t h e r x E I o r x* E I. Proof.
(iii): If x E I then 'plx E I hence x* = Cplz # I because I is proper. If x # I then 'plx# I hence the prime ideal I f l C ( L ) of C ( L ) (cf. the dual of Lemma 2.2) contains Cplx (6. t h e dual of Proposition 1.3.18).
(i)
j
(iii)
+ (ii):
contradiction.
If x E I n D s ( L ) then x E I , z* = 0 and x*
# I,
a
Prime filters
269
l(i) =+ l(ii): If z E I and cp1z (i!I then as for (i) + (iii) we get Cplz E I n C ( L ) , hence z V Cplz E I and since cpl(z V plz) = 1 it follows that z V qlzE I n Ds(L).
0
2.13. Remark. In this book there are several uses of t h e symbol Ker which should not be confused, even if they are not distinguished by subscripts. The first meaning is the one used in universal algebra: if A, A' are r-algebras then
(2.2)
kerA,At
: Hom(A,A') + @(A)
is defined as in (1.5.6), i.e. for every f E Hom(A,A'), (2.3)
*
~(ker~ f , )~ ~ ) f(z) = f
( ~ )( V ~ , YE A ) .
The first occurrence of this concept was in formula (1.3.8), in t h e particular case of lattices. As a matter of fact we are interested in t h e more particular case of algebras L , (2.4)
kerLp
L' from LM9 or LMNd. Note that in this case
: Hom(L,
L')
-+
Oo(L)
because if p;z(kerL,Lt f)p,y (Vi E I) then cpif(4
= f ( c p i 4 = f(cpiY) = Vif(Y)
(Vi E I)
therefore f(z)= f(y) i.e. z(kerL,Lt f)y. The next meaning of ker was first used in Theorem 1.13, again for L , L' E LM19 or LMNB: (1.4)
kerL : Oo(L)-+ F16(L)
,
(1.7)
ker p = { z E L I z p l }
(Vp E O O ( L ) )
A third and similar meaning will be introduced now. 2.14. Theorem (Cignoli [1969a], Georgescu [1971d]).
Let C = LM19 or C = LMNB. For every L E C there is a bijection between PFld(L) and Homc(L, L;'). Proof. Set Homc(L,L, UI) = H and KerL = kerL o k e r L , y i.e., for each
f EH,
Filters, ideals and d-congruences
270
K e u f = 1. E L I s(kerLLrJl ' 2 f)ll = f-1([1))
(2.5)
7
l(i) = 1 (Vi E I). As L;' is a chain by Lemma 4.6.11it follows that [l) is a prime d-filter
where 1 E Li',
of Li', hence it is straightforward t o check that f-'([l))is a prime 6-filter of L. Thus KerL :
H + PFlO(L).
Now we prove the existence of KerL', namely
(Keri'F)(x)(i) =
(2.6)
(
1
,
vix E F
0
7
vix $ F
(Vx E L) (Vi E I ) ,
for each F E PF16(L). Let f ( x ) ( i ) stand for the right side of
f(x) E ' iL
(Vx E L).
The next step is t o check that every
(2.6).Clearly
f
E
H . Using (3.1.11) we
see that for
x E L and i,j E I,
* Yjcpix E F H Fix E F =1* (f(x))) (.i> =1
f(vjx)(j) = 1 H
f(
(vi
e
7
proving that f(cpiz) = cpi ( f ( z ) ) .One verifies similarly, using Remark 1.3.5.
1.3.9,that f i s a bounded- lattice homomorphism. In view of Remark 3.1.29 (and of Remark 1.2.29 if C = LMNd) this shows that f E H . Then using Remark 1.2 we obtain a) and Proposition
x E ker f
* f(x) = 1 H f(z)(i) = 1
H cpix~F
satisfies
I) e
(V~EI)HZ€F,
f = F . Finally it remains ker g = F then g = f. This follows from
which proves t h a t ker
(Vi E
t o show that if g E H
271
Prime filters 2.15. Corollary (Georgescu [1971d]). If L E P19 t h e n a) every m o r p h i s m f :
b) L'\
L +L '\
is a surjection, and
has n o proper Post subalgebras.
Proof.
ra
(C(L)) , hence Proposition 4.1.6 yields an element x E L such that 'pix = u ( i ) (Vi E I). Notice that f = ker-' ker f by Theorem 2.14 and f ( u ( i ) ) = 1 or 0
a) Take a E
Lia. But
a can be viewed as a funtion a E
according as u ( i ) = 1or 0. Then f(x) = a because
f(z)(i) = 1 H 'pix f ker f H u(i) E ker f
* f (a(;)) = 1
($
a(;) = 1 ,
b) Apply a) t o the inclusion mapping of a subalgebra L of LiA.
0
2.16. Corollary.
T h e following conditions are equivalent
L E P19 and f E F119(L):
~ O T
(i) F is a maximal 19-filter;
(ii) L / F Proof.
(i) j ( i i ) : The morphism f = Ker-'F : L + is a surjection by Corollary 2.15, therefore L'\ FZ L / ker f by the homomorphism theorem 1.5.16. Thus it remains to prove that ker f = modF. First we apply (2.6) and obtain
Now if z ker f y then for each i E I, (2.7) and Theorem 2.3 imply that either ( p i x , (piy E F in which case we set uj = 'pixA (PigI or (pix, Cpiy E F in which case we set u;= pix A p;y. In both cases
Filters, ideals and 29-congruences
272
that is z m o d F y . Conversely, (2.8) implies that for each i E I,if via: E F then (piy A ui E F hence (pig E F and similarly (p;y E F + pix E F therefore z ker f y by (2.7).
(ii) + (i): Suppose F’ is a proper 29-filter of L such that F E F‘. Then = {2 I z E F’} is a proper &filter of L/F by Proposition 1.19. As 0 remarked after Definition 1.1, this implies F? = {i},that is F‘ = F .
2.17. Proposition. T h e following conditions are equivalent f o r L E LM29 and F E Fl(L):
(i) F is mazimal; (ii) there is a p r i m e filter G of C(L) (necessarily unique: G = F’) such that F = GI. Proof.
(i) + (ii): In view of Lemma 2.2 we obtain in turn that F* is prime in C ( L ) , F C (F*)1 and (F*)l is prime in L , hence proper; therefore F = ( F * ) l . If F = G1then F’ = (Gl)* = G by Lemma 2.2. (ii) =$ (i): If F’ is a proper filter of L such that F F’ then G = F’ (F’)’ C C(L) hence F’ = (F‘)’ therefore F = F’ by Lemma 2.2. 0 2.18. CorolI a ry . If L E LM29 and F E PFl(L) then F’ = (F*)oand F” = ( F * ) l are the unique m i n i m a l p r i m e filter F’ F and the unique m a x i m a l filter F” 2 F , respectively.
s
Proof. In view of Lemma 2.2, F’ is a maximal filter of C(L) and F‘ C F E F”. But F’ is a maximal &filter by Theorem 1.7, therefore a minimal prime filter by Theorem 2.3, while F” is a maximal filter by Proposition 2.17. If P is a maximal filter such that F P then F* & P” and F’ E F”’, therefore P” = F’ = F” hence z E P implies (pox E P’ = F”’ F” then z E F”, which shows that P E F”, therefore P = F”. The uniqueness of F’ is
s
273
Prime filters proved analogously.
A. Monteiro [1974]has shown that for L E D01, every prime filter is included in a unique maximal filter if and only if L is n o r m a l i.e. for every x,y E L satisfying x Ay = 0, there exist u , v E L such that X A V = y A u = 0 and u V v = 1. Then it follows from Corollary 2.18 t h a t every L E LM9 is normal; this can be also proved directly taking u = 'plyVx and v = cplxVy.
2.19. Definitio n. Let L E LM9. Denote by Rad(L) the intersection of all maximal filters of L. For every proper filter F of L let Rad(F) stand for t h e intersection of all maximal filters of L that include F . 2.20. Proposition. Let L E LM19 and set M = PFl(C(L)). Then
(2.9)
Rad(L) =
FEM
Fl = Ds(L) .
Proof. The first equality follows from Proposition
Rad(L) = 9';
( FEM n
F ) = cp~'([l))
2.17,implying further
= Ds(L) .
2.21. Proposition. Let L E LM9 and F a proper filter of L. Then Rad(F) = cpT'(F). Proof. Let
MI be the family
L which include F that M1 = {G1 I G E
of those maximal filters of
M z = { G E PFl(C(L)) I F* G G } ; we claim M z } . If F' E M1 then F' C F", while Proposition 2.17 shows that F' = ( F f * ) land F'* E PFl(C(L)), hence F'* E M2. Conversely, suppose F' = G1 where G f M 2 . Then F' is maximal by Proposition 2.17 and 2 E F implies cplx E F* G G hence x E GI = F', proving that F 5 F', therefore F' E M I . Now using Corollary 1.3.14in C(L) for F* we obtain and
Rad(F) =
n
GEM2
cp;l(G)=cp;l(
n
GEM2
G)
=
Filters, ideals and 19-congruences
274
The next result connects filters t o the operations Ch. 4, $3.
and
% studied in
2.22. Theorem. The following conditions are equivalent f o r L E LMn: (i)
L is a Boolean algebra;
(ii) Rad(L) = [l);
(iii) every proper filter F is a n intersection of maximal filters, i.e. Rad(F) = F ; (iv)
x
+ y = ( P ~ - v~ y;x
Proof.
(i) + (iii): By Corollary 1.3.14 and Proposition 1.3.18. (iii) + (ii): In particular Rad([l)) = [l); but clearly Rad([l)) = Rad(L ) . (ii) + (i): As x A x* = 0, it remains t o prove that x V x* = 1. But x V x* E D s ( L ) , because (x V x*)* = 0 and D s ( L ) = [l) by Proposition 2.20. (i) + (vi): By Proposition 4.3.6. (vi) + (v): By (4.3.25) and (3.1.27) where 0 is 1. (v)
+ (i): The
(1
+ x) V (x +- 0)
= 1, that is : ( p ; ~ V c p ~= - ~1,xi.e. ( P ~ - 5 ~ X(pix.
hypothesis yields
xVcp,-lx = 1, whence for each i c I Thus (pix = (P,-~X (Vi E I ) , which shows that L (i) + (iv): By Proposition 4.3.6.
= C ( L ) E B.
275
Prime filters (iv)
+ (i): x* V x = qn-1x V I = x =$ x = 1.
0
2.23. Remark. Properties (i)-(iii) in Theorem 2.22 are equivalent even for L E LM6 (same proof).
A. Monteiro (cf. A. Monteiro [1974]) raised the problem of whether a bounded distributive lattice in which every proper filter is an intersection of maximal filters and dually, is a Boolean algebra. The answer is negative, as was shown independently by Balbes [1972] and Adams [1974]. However for
L E LMn the answer is positive; cf. Theorem 2.22. 2.24. Proposition.
The fol2owing conditions are equivalent for L
E LMt9 and
F E PFl(L):
(i) F is maximal; (ii) D s ( L ) Proof.
c F.
*
(i) (ii): By Propositions 2.19 and 2.16. (ii) =$ (i): Let F' be a filter such that F c F'. Take 5 E F' - F. From yl(x V p1x) = 1 we infer I V Flx E D s ( L ) 5 F and since F is prime it 13 follows that $311 E F 5 F', therefore 0 = I h Cplx E F', i.e. F' = L.
In the case of LMn-algebras we can say much more about prime filters. 2.25. Theorem (Boicescu [1984]). T h e following conditions are equivalent f o r L E LMn and a proper filter F of L:
(i)
F is prime;
(ii) there is a prime filter
F' F ;
(iii) ihere is a unique minimal prime filter F'
F;
276
Fiiters, ideals and 6-congruences
(iv) F* is prime in C ( L ) ; (v)
there is a prime filter G o f C ( L ) (necessarily unique: G = F*) and i E (1, ...,n - 1) such that F = G;;
(vi) F is included in a unique maximalfilter. Proof.
(i) j (iii): By Corollary 2.17. (iii) + (ii). Trivial. (ii) +- (iv): F’* E PFl(C(L)) by Lemma 2.2 and F’* = F*.
F“
C_ F*, therefore
c
F (F*)n-l by Lemma 2.2. We prove that F = (F*)jfor j = max {il(F*)iC F } . If j = n-1 then (F*),-lC F hence F = (F*)n-l. If j < n - 1suppose by way of contradiction the existence of x E F - (F*)j. Then ‘pjx @ F’ hence cpp E F*. But (F*)j+l F hence there is y E (F*)j+l such that y @ F . Using Proposition 3.2.3we obtain y 2 x A Cpjx A ( ~ j + E ~ yF therefore y E F , a contradiction. If F = Gi for G E PFl(C(L)) then G = F’ by Lemma 2.2. (v) +- (i): By Lemma 2.2. (i) 3 (vi): By Corollary 2.17. (vi) + (v): F is an intersection of prime filters F’ by Corollary 1.3.14. Each F’ is included in a maximal filter P by Corollary 2.18. But F 5 P therefore P is unique by hypothesis. Then P* and each F” are maximal in C(L)by Lemma 2.2 and also F” P*,therefore F‘* = P* for each F‘. According t o (iv) 3 (v), for each F‘ there i s a j such that F‘ = ( P * ) j ,which implies F = ( P ” ) k ,where k is th e least j. This representation is unique with 0 respect t o P* again by Lemma 2.2. (iv) 3 (v):
(F*),
c
Comment . Theorem
2.25 shows that if 6 = n certain properties of &filters can be
(iii) extended t o filters. Thus the equivalence (i) ($ (iv) generalizes (iv) in Theorem 2.3; (i) ($ (iii) and (i) (vi) generalize Corollary 2.18 and the lattices sharing this property have been characterized by A. Monteiro
[1974]. In that paper it
is also proved that for
L
E
D01,every proper filter
277
Prime filters
which includes a prime filter is also prime if and only if L is fully normal; cf. Definition 4.3.14. The above equivalence (i) e (ii) implies that every LMn-algebra is fully normal; cf. Proposition 4.3.16. The equivalence (i) ($ (v) generalizes a result of Cignoli [1969a] on Moisil n-algebras (see also Cignoli [1975a]). 2.26. Corollary (Cignoli [1969a], [1975a]). If L E LMn then t h e poset PFl(L) is the cardinal s u m of a set of chains of cardinality 5 n - 1 each: (2.10)
{G; I i = 1, ...,n - 1)
, G E PFl(C(L)) .
Proof. Taking into account Theorem 2.25 it only remains to prove that every two elements belonging to distinct chains (2.10) are incomparable. If Gi C Gi for G,G‘E PFl(C(L)) then G = (G;)’ (G>)* = G‘, therefore
G = G‘.
0
2.27. Corollary (Epstein [1960], Traczyk [1963]). I f P E Pn then each chain (2.10) has exactly n - 1 elements. Proof. For each G E PFl(C(L)) and each i = 2, ...,n - 1, from (pic,-i = 1 E G and ( P ; - ~ C , - ~ = 0 @ G we infer c,-i E G; but c,-i @ Gi-1, therefore
Gi-1
c Gi.
0
The converse of the above result will be the object of the next section 2.28. Corollary. If L E LMn, F’,F” E PFl(L) and F’, F” are incomparable t h e n F’VF” = L. Proof. If F’V F” is a proper filter then there is F E PFl(L) such that F’VF” F , which implies F” C F* and F”’ E F’ hence F” = F” = F’”, therefore Theorem 2.25 yields F‘ = (F*)i and F” = (F*)j for some i , j . 0 This contradicts the incomparability of F’and F”.
278
Filters, ideds and d-congruences
2.29. Proposition. If L E LMn is infinite then card L
5 card PFln(L) = card PFl(C(L)) ,
Proof. It follows from Remark 2.4.c and Corollary 2.26 that (2.11)
cardPFl(C(L)) = cardPFln(L) 5 cardPFl(L)
5
5 ( n - 1)cardPFI(C(L)) . But C ( L )is infinite by Proposition 1.25, hence cardC(L) 5 cardPFl(C(L)) (Makinson [1969]), therefore PFl(C(L)) is infinite and (n-l)cardPFl(C(L))
= cardPFl(C(L)) which together with (2.11) yield
cardPFl(L) = cardPFl(C(L)), while cardL Makinson result.
5 cardPFl(L)
again by the 0
Proposition 2.29 cannot be generalized to infinite 6; thus e.g. Liq is infinite while cardPF1d(Lhq) = 1 as noticed after Remark 1.2. 2.30. Remark. The theory presented in this section can be extended in several ways. One of them has been already mentioned: the passage from LM-algebras to bounded distributive lattices. Another idea is followed in Sicoe (19711: the study of the ideals of the form c p i ( K ) or cp;l(IC), where i E I is fixed and I< runs over the ideals of L. lorgulescu [1984c] has obtained a natural generalization of Theorem 2.3 and several related results within the framework of m-concepts; cf. Remarks 1.3.31 and 1.20. In particular it is noted that every maximal 9-filter which is also an m-filter is obviously a maximal (m, d)-filter but it is not known whether these two concepts coincide and their relationship to the concept of maximal 19-filter is also an open problem. lorgulescu [1984c] also proved that in the d-valued algebras (d-infinite) the equivalence (i) % (v) in Theorem 2.25 does not hold.
The Cignoli problem
279
53. The Cignoli problem Epstein [1960] and Traczyk [1963] proved t h a t i n every Post algebra following property
L the
P holds: th e set PFl(L) o f prime filters of L is a cardinal
sum of ( n - 1)-element chains. In his thesis, starting f r o m Moisil algebras, Cignoli gave a new proof o f the above theorem (cf. Corollary 2.26) and asked whether the converse was true. More precisely: is every Moisil n-algebra w i t h
P a Post algebra? In this section we prove t h a t the answer is affirmative even in the more general case when L E LMn; cf. Boicescu [1984]. property
3.1. Lemma.
Let L E LMn, with property P . If F E PFl(C(L)) let Fl
c F2 c ... c
be the chain of prime filters in L determined by F (cf. L e m m a 2.2). FOT each i = 2, ...,n - 1, there exists x b E L, such that:
F,-1
(i) x & E ~ - - - * ( i = 2 ,...,n-I),
(iii) 'pix> =
= ... = ' p n - l x $ .
3.2. Lemma.
Let B be a n n-valued Post algebra and F E PFl(C(B)) and i E ( 2 , ...,n1). T h e n the elements
satisfy conditions (i)-(iii) of L e m m a 3.1. Conversely, every element 2; satisfying conditions (i)-(iii) of L e m m a 3.1 has a unique representation
Filters, ideals and &congruences
280 of the f o r m (3.1), n a m e l y yk = 'pix.
Proof. It follows from (3.1) that 'pjx; = 0 for all j < i and 'pix; = y$ for all j 2 i. Therefore (ii) and (iii) hold. In particular ' p i - 1 ~ ; = 0 hence x; $ F,-1, while 'pixi = yk E F hence 3% E F;. Thus (i) holds and the representation (3.1) is unique. Conversely, if Z> = cn-i
&EB
satisfies conditions (i)-(iii) of Lemma 3.1, then
A 'pix; foIIows from
3.3. Lemma. Under the hypotheses of L e m m a 3.2, there exists a finite subset 'Po C PFl(C(B)) such that
Proof. In view of Proposition 1.3.9, the elements of SpecC(B) are the sets L - F , F E PFl(C(B)). But y> g! L - F hence L - F E s(y$) (cf. Notation 2.1.5) for each F E PFl(C(B)), therefore SpecC(B) = U{s(yk) I F E PFl(C(B))}. Now the compactness of SpecC(B) and Corollary 2.1.6 imply the existence of a finite subset Po S PFl(C(B)) such that
whence (3.2) follows by injectivity of s.
0
3.4. Theorem. If L E LMn and satisfies condition P , t h e n L i s a n n-valued Post afgebra. Considering the canonical monomorphism (4.1.3) F'L : L + C(L)["-'I, it follows that L can be identified with the LMn-subalgebra PL(L) Proof.
The Cignoli problem
281
C(L)["-l]and it is sufficient to prove the theorem for PL(L). As C(L)["-'] is a Post algebra and therefore has property P , it follows that for any F E PFl(C(C(L)"+'])), the chain of prime filters l increasing. As in C(L)"+'], ( ~ ) l ~ i ~ nis- strictly
of the n-valued Post algebra
(3.3)
C(PL(L))= C(C(L)["-']),
P,it follows that the chain of prime filters in PL(L) F1 n PL(L) c FZ n PL(L) C ... C Fn-1 n PL(L).
and PL(L)has property
F
determined by
is
According t o Lemma 3.1, there exists z& E PL(L) satisfying conditions
(i)-(iii). However zk belongs t o the Post algebra C(L)"+'] and therefore by applying Lemma 3.2, it follows that there exists y b E F , such t h a t zb = c,-i A yk, for any i = 2, ...,n - 1 and any F E PFl(C(PL(L))). According t o Lemma 3.3 there exists a finite subset Po c PFl(C(PL(L)))
V
such t h a t
yk = 1. But in
C(L)fn-']
FEPo
and because
V
FEPo
z& E PL(L),it follows that c,-; E PL(L)i = 2, ...,n-1.
Using again (3.3) we see that PL(L) is an n-valued Post algebra.
0
Comment.
The above proof implies in fact (3.5)
4;n-i
= V{z$ I F E PFI(C(PL(L)))}
(i = 2, ...,n - 1)
2 z$ for all F and if z 2 zb (VF) then x 2 c,-i by (3.4). As a matter of fact (3.5) holds in PL(L)because cn-i E PL(L). Conversely, a direct proof of (3.5) i.e., for L and without using Lemma 3.2 and 3.3because c,-i
would yield a direct proof of Theorem 3.4-i.e.
without using Post algebras
or topological arguments. We are able t o give such a proof in the case when
L is complete.
282
Filters, ideals and 19-congruences
3.5. Proposition. Let L E LMn be complete and with property P. Then (3.6)
( i = 2, ...,n - 1) ,
V { x b I F E PFl(C(L))} = cn-;
where x f . are the elements constructed a'n Lemma 3.1 and cn-i satisfy (4.1.9)(cf. C o r o k l ~ y4.1.9). Proof. We adopt (3.6) as a notation and prove (4.1.9), i.e. cpjc,-i = 1or 0 according as j
Let i E (2,
2 i or j < i. It suffices t o prove ( P ; c , - ~= 1and ( ~ i - ~ c = ~ -0.i
...,n - 1).
F E PFl(C(L)), from cn-i 2 x ; and x ; E F; we obtain (Pic,-; 2 (Pix; E F , hence Propositions 2.7 and 1.3.18 imply For every
(Pic,-i
E
n {F I F E PFl(C(L))} = (1) .
Now suppose by way of contradiction that exists F E PFl(C(L)) such that follows t h a t
~ p i - ~ A ~ -cpix; i
we infer x;,
5 pix;
E
(Pi-1Cn-i
which contradicts (3.7).
#
(~i-~c,-i
0. Then there (pix& E F it
F and since
F ,hence
V x ; . Therefore cn-i
turn
E
5 q ; x & V 2% and this
implies in
The Cignoli problem
283
We reminded above t h a t if L E DO1 and L is an n-valued Post algebra, then L has property P . The converse is not true, as was proved by Chang and Horn [1961]. They have considered L = L:, where S = {sili E w @ 1). Obviously L E DO1 and let L1 be the subset of L formed by the elements such t h a t f ( s n ) = c2, for any n E w, n 2 n o , for some no E w and f(sw) = c1 or c2, or f ( s n ) = co, for any n E w, n 2 no, for some noE w and f(sw) = co. One proves that L1is a sublattice with 0 and 1of the lattice L and has property P , but L1 is not a three-valued Post algebra. One can say even more:
3.6. Proposition. L1 E DO1 defined above has no structure of three-valued LukasiewiczMoisil algebra. Proof. As
L1 satisfies condition P
but cannot be a three-valued Post alge-
bra, this follows from Theorem 3.4.
0
This proposition was initially proved by Cignoli [1969b]. Chang and Horn [1961] and then Traczyk [1963] gave necessary and sufficient conditions for a bounded distributive lattice having property P t o be a Post algebra, but in those characterizations appear the constants ci from the definition of a Post algebra.
3.7. Corollary. If L E DO1 and satisfies property P , then the following are equivalent:
(i) L is a n n-valued Post algebra;
(ii) L has a structure of n-valued Lukasiewicz-Moisil algebra. 3.8. Corollary. If L E DO1 and satisfies property P , then L has a unique structure of n-valued Lukasiewiez-Moisil algebra compatible with the lattice structure. Proof. By Theorem 3.4 it follows that L has a structure of n-valued Post algebra. This structure is unique (cf. Balbes and Dwinger [1974] p. 192
284
Filters, ideals and &congruences
or Exercise X.2.8) and it determines uniquely the structure of LukasiewiczMoisil algebra; cf.
(4.1.11)and Corollary 4.1.9.
0
3.9. Corollarv.
If L is a n n-valued P o s t algebra then L has a unique structure of n-valued Lukasiewicz-Moisil algebra compatible with t h e lattice structure of L.
3.10.Corollary. If L E DO1 and PFl(L) as a cardinal s u m of chains of cardinality 2, t h e n the following are equivalent:
L i s a double pseudocomplemented lattace;
(i)
(ii) L E M3; (iii) L is a three-valued Post algebra; (iv) L is a double Stone algebra; (v)
L i s a double pseudocomplemented lattice and D s ( L )n
# 0.
Proof.
(i)
by Varlet [1968]. (ii) + (iii): By Theorem 3.4. (iii) + (iv): By Theorem 4.4.4. (iv)
(iii)
=$-
(ii): Proved
+ (i): Obvious. + (v): There exists c1 E L , such that (plcl= 0 and cpzcl = 1. By
Proposition (v)
4.4.6,c1 E D s ( L ) n
(i): Obvious.
m.
0
CHAPT E R 5
FILTERS, IDEALS AND 8-CONGRUENCES
T he aim of this chapter is t o construct a theory of filters, ideals and congruences in LM-algebras generalizing the main results of t h e correspon-
ding theory for distributive lattices (6.
Ch. 1, 93).
T h e starting point is
a convenient definition o f th e concepts of filter, ideal and congruence in LM-algebras, obtained by adding appropriate extra conditions to the corresponding concepts of lattice theory and universal algebra, respectively (cf.
Ch. 1). T he last section answers a problem raised by Cignoli i n connection wit h a property o f prime filters in Post algebras.
$1.d-filters, d-ideals and 8-congruences
A 9-filter (&ideal) of an LM-algebra is quite naturally defined as a filter (an ideal) compatible w i th the endomorphisms cpi. It turns o u t t h a t &filters coincide wit h Stone filters and w i th deductive systems. Since the quotient
L / p of an LM-algebra L by a congruence p (in the sense of universal algebra) need not satisfy th e determination principle, we introduce t h e stronger concept of 9-congruence. T h e “metatheorem” of this section is t h a t 6- filters (&ideals),
d-congruences and quotient algebras behave as one would
expect. See also Hatvany [1984].
1.1. Definition. Let L E LMd. A d-filter of L is a filter F of L such t h a t A d-ideal of L is an ideal H of
x E F jcpox L such th a t x E H + cp1z E H .
E
F.
In the sequel we only deal w i th &filters b u t all the results can be transferred by duality to 1’1-ideals.
Filters, ideals and 6-congruences
248
1.2. Remark. For every &filter
F
The concept of &filter was introduced by Moisil [1940], [1963c] for nalgebras and later [1968] for d-algebras, under the name of strong filter (because there exist filters that are not &filters, e.g.
F
#
(1) of L d : if x E F - (1) then pox = 0
proper d-filter of
Liq is (1)). However,
#
every proper filter
F ; similarly the unique
it turns out t h a t in the case of
LM-algebras 6-filters coincide with Stone filters introduced by A. Monteiro
[1954] (6. A. Monteiro [1974]) and studied by Cignoli [1966a], [1966b] and Cherciu [1971].
1.3. Definition.
A Stone filter of a bounded (distributive) lattice L is a filter F of L such that for every x E F there is y E F n C ( L ) such t h a t y _< z. We set
F* = F n C ( L ) .
(1.1)
1.4. Proposition.
The following conditions are equivalent for a filter F of an algebra L E LM6: (i) F is a 6-filter; (ii) F is
a
Stone filter;
(iii) F is generated by F*; (iv) Proof.
F = cp;l(F*).
+ (ii): For x E F take y = pas. + (iii): Obvious. + (iv): Because z E F * pox E F H 'pox E F*
(i) (ii) (iii)
249
&filters, 19-ideals and 19-congruences (iv) =+ (i). Because x E
F =+ 'pox E F' G F.
Comment.
It is easily seen that F* is the unique filter of C(L) satisfying (iv).
1.5. Definition. Let L E LM9. For every subset X of L we denote by [ X ) , the &filter generated by X , i.e. t h e least 19-filter including X . Also, we denote by Fl(L) and F119(L) the lattice of all filters of L and t h e lattice of all 19-filters of L, respectively.
1.6. Remark. Let L E LM19. Then [0)4 = (1) while if (1.2)
[XIS
= {Y E L I 'po(zi A
I
0# X
L then
... A xn> I
y;sl,...,2n~~;n~mr-{0}};
C C ( L ) then [ X ) , = [ X ) while if X is a filter of L then [X>S= {Y E L I 'pox I y, 2 E X } .
in particulx if X
1.7. Theorem.
For every L E LM19 the m a p s * : F119(L) + Fl(C(L)) defined by (1.1) and 'pi' : Fl(C(L)) -+ F119(L) are lattice isomorphisms anverse t o each other. Proof. Clearly the maps * and 'pol are isotone and their targets are those indicated above. The composites
'pi'
o * and * o 'pi' are the identity
mappings by Proposition 1.4 and
g,'(P) n C(L) = P which follows from x E
( V P E FlC(L)) ,
C(L)n 'p;'(P) -s x = 'pox E P -s 5 E P .
The congruences in the classes LMQ and LMNQ are introduced according t o the general Definition 1.5.11. However, compatibility with 9; and N follows from the other conditions:
Filters, ideals and 8-congruences
250
1.8. Proposition (Boicescu [1984]). T h e following conditions are equivalent f o r a n equivalence relation p of a n algebra L E L M 8 :
(i) p is a congruence; (ii) p is compatible with A, V and
'p;
(Vi E I ) .
Proof.
(i) j (ii): Trivial. (ii) + (i): We suppose x p y and prove CpixpCpiy. But ' p i x p p i y hence 1p('piy V p i x ) and similarly 1p ('pix V Cpiy). This implies in turn
and similarly piypCp;x A Cpiy, therefore p i x p p i y
(Vi E I ) .
0
The construction of quotient algebra requires a stronger concept (see Remark 1.16). 1.9. Definition (Boicescu [1984], lorgulescu [1984c]).
A 29-congruence of an algebra L E L M 8 or L E LMN29 is a congruence of L E DO1 such that x p y ('pix p (piy (Vi E I ) ) . 1.10. Proposition.
T h e following conditions are equivalent f o r a binary relation p of a n algebra L E LM29;
(i) p is a 29-congruence; (ii) there is a congruence p' of L such that z p y Proof.
(i) + (ii): Take p' = p.
@
(Pix p ' v i y (Vi E I ) ) .
d-filters, 19-ideals and &congruences
251
(ii) + (i): Clearly p is a congruence and if 'pjxpcpjy (Vj) then 0
cpicpjx p'(pj(pjy ( V i , j ) , therefore (pjx p'(pjy (Vj), i.e. z py.
1.11. Definition. Let L E LM29 or
L E LMN29. For every congruence p of L we denote by
p the d-congruence
generated by p , i.e. the least d-congruence including p.
O ( L ) and Oo(L) the lattice of all congruences of L and the lattice of all &congruences of L , respectively. Also, we denote by
1.12. Remark. Let L E LMr9 or L E LMNd and p E z P Y @ (piz ppiy
(1.3)
O ( L ) . Then
(vi E 1))
*
To establish t h e relationship between d-filters and d-congruences we intraduce the maps kerl;, modl; and modl;, or simply ker, mod and mod: (1.4)
ker : Oo(L) --t F119(L) ,
(1.5)
mod : F119(L) --t O ( L ) ,
(1.6)
mod
: Fld(L) + Oo(L)
defined by (1.7) (1.8) (1.9)
ker p = {z E L I z p l }
(Vp E Oo(L)) ,
z m o d F z ' @ 3u E F z A u = z' A u
- mod =
o mod
(VF E Fld(L))
,
(cf. Definition 1.11) ,
respectively.
1.13. Theorem (Boicescu [1984],lorgulescu [1984c]). Let L E LM19 OT L E LMN.9. The maps mod and ker satisfy mod = ker-' and establish an isomorphism F129(L) S Oo(L). Proof. It is easy t o check that ker p E Fld(L) even for p E O(L); e.g. if z E ker p and z
5 y then y = z V y p 1V y = 1i.e.
y E ker p. Let us prove
252
Filters, ideals and 19-congruences
that m o d F E O ( L ) for F E F119(L). Take x m o d F x ' a n d y m o d F y ' i.e. x Au = x'Au a n d y A v = y ' A v for u,v E F. Then (PiXAviu = (PiYAviu where (piu E F, hence 'pix mod F'piy, while from x A u A v = x' A u A v and y A u A v = y' A u A v where u A v E F, we deduce immediately that ~ A y p ~ ' A y ' a nxVypx'Vy'. d If L E LMN6 then N x V N u = Nx'VNu. But from
we get N u A 'pou = 0 hence
NXA YOU = ( N x V N U )A C ~ O U = ( N d V N U ) A
YOU
=
where pou E F, showing that Nxmod F Nx'. Clearly the maps ker, mod and are increasing. To prove ker o m o d = 1 F l q ~ take ) F E Fld(L). Then
-
-
ker(modF) = {x E L zmodFI} = = { X E L vix mod F 1 (Vi E
I)}
=
= {XEL
by Remark 1.2. To prove mod o ker = 1e0(,qnote first that you 5 u and ker p n C(L)= ker(p n (C(L))')and recall Proposition 1.3.26. Then for every p E Oo(L)
zmod
o ker p y H pixmod ker pviy
(Vi E I)
19-filters, 19-ideals and d-congruences
253
1.14. Proposition (Boicescu [1984]). Let L E LMn or L E Mn. T h e n Oo(L)= O ( L ) and the maps mod and ker satisfy mod = ker-' and establish a n isomorphism Fln(L) O ( L ) . Proof. Let p E
-
O ( L ) . If x mod o ker p y then (i = 1,...,n - 1)
cpixmod o ker pcp;y
w (3u; u ; p l & cpix A ui = (piy A ui) therefore taking u = u1 A
(i = 1,...,n - 1) hence
... A u,-'
(i = 1, ...,n - 1)
we obtain u p 1 and cpix A u = (piy A u
(i = 1,...,n - 1) , therefore x A u = y A u, showing that
mod o ker p therefore
(1.10)
3 mod
-
o ker p y. Thus mod o ker p
E
-
mod o ker p = m o d o ker p E Oo(L) .
Taking into account Theorem 1.13 and (1.10) we have
-
xpy+xpy*xmod (j
o kerpy
cpixmod o ker pcp;y
w
(i = 1, ...,n - 1) e
* (3% u; p 1 & pix A u; = cp;y A u;)
(2
= 1,...,n - 1)
+
+- (3% (o;u~p 1 & c p ; ~A ~piui= (P;Y A 9;";)
(i = 1, ...,n - 1)
+ (i = 1,...,n - 1)
+
=$
(pix mod o ker py;y
=$
xmodokerpyw3uupl&xAu=yAu
+- 3~ x = x A I p s A U = Y
A U ~ YA 1 = y
proving that p = mod o ker p, therefore p
O(L)= Oo(L), hence by Theorem 1.13.
-
+. X
+ ~
Y
E Oo(L)by (1.10). This proves
is the identity mapping and the proof is complete 0
254
Filters, ideals and d-congruences
1.15. Corollary.
I f B is a Boolean algebra t h e n Oo(B)= O ( B ) . F12(B) = FI(B) and the m a p s mod and ker satisfy mod = ker-' and establish a n i s o m o r p h i s m Fl(B) O ( B ) . 1.16. Remark. Definition 1.9 ensures that if p is a 29-congruence of an algebra L E LM6 ( L E LMN29) then the determination principle holds in L / p , therefore L / p E LM19 ( L / p E LMNB). In view of Theorem 1.13 every &congruence p is of t he form mod F for some &filter F (= ker p). The quotient 29algebra L / m o d F will be denoted simply by L / F . In particular if L E LMn or L E Mn then every n- congruence is of t h e form m o d F and L / F will stand for L/mod F .
1.17. Remarks. a) In a Moisil n-algebra Proposition 1.8 remains true: If L E Mn we still
have t o prove that x p y implies N x p N y . This follows from
Therefore a l l types o f congruences reduce t o lattice congruences compatible with 'pi.
b) However, this does not hold in general, as shown e.g. by the following example due t o lorgulescu 11984~1.Suppse I is an infinite regular set (6. Definition 4.6.7) and take L = (?(I))['. Let Fo C C ( L ) be the filter of those functions f E C ( L )for which the constant value f ( i ) is a cofinite set (i.e., I - f ( i ) is finite). We denote by F the &filter generated by Fo and prove that mod F is not a &congruence. Let g, h E L be defined by g ( i ) = [O,i-]C I for i # 0, g(0) = 0 and h ( i ) = [O,i]G I,for Vi E I . For each i E I, define f; E C(L)by f i ( j ) = I - {i} V j E I. Then f i E FO C F and ('pig
fi)(j)
= g ( i ) n f i ( j ) = [O, i-]
n ( I - {i})
= [o, i-]
&filters, 19-ideals and d-congruences
255
and similarly ((pjhA f i ) ( j ) = [0,i-1, therefore (pig A fi = (pihA fi, proving that YigmodFcpih for every i E I . It remains to show that ( g m o d F h )
does not hold, i.e. that g A
f # hAf
for every
f E F.
f E F fo 5 f.
But
implies, via Remark 1.6, the existence of fo E Fo such that
This implies further the existence of a finite subset loc I such that 8 # I - I0 = fo(i) 5 f(i) Vi E I. Take i E I - Io; then g A f # h A f follows from
1.18. Remark.
It follows from Remark 1.17 b) and Proposition 1.8 that for certain sets 1 the corresponding classes LMd and LMNd are not equational (because the homomorphic image L / p of L is outside t h e class whenever the congruence p is not a &congruence).
1.19. Proposition. Let L E LMd or L E LMNd. If F, F’ E Fld(L) and F 5 F‘ then = {i I x E F’} is a 29- filter of L / F . If, moreover, F’ is proper (prime) is so. then
F?
Proof. The first statement is obvious. If 0 E
u; E F (Vi E I ) , hence 0 =
(pis A
then 0 = ( p i x A ui, s E F’,
(p;ui, which implies (piu; 2 p i x E F‘
F‘ and since (p;ui E F C_ F’ it follows t h a t 0 E F’. If 3 V E F? then pi(^ V y) A ui = (pi2 A U i , z E F‘, ui E F (Vi E I ) , hence ( p i x A U i ) V ( v i y A U i ) E F’, implying e.g. (pox A uo E F’, in which case x E F‘ because x 2 (pox A ug, therefore i E F? otherwise 6 E F?. O
therefore piui E
1.20. Remark (lorgulescu [1984c]). The concept of d-filter and th e concept of m-filter in Remark 1.3.31 are independent of each other. Thus e.g. a) if
I
23 then the filter I of Ld = J = (0) + I is neither a &filter nor an m-filter. Further b) let E be a set of cardinality m and the filter F of P(B)consisting of all cofinite sets.
Filters, ideals and &congruences
256 Then E - {z} E
F
for every z E
E and
n
(E- {z})
xEE
= 8 !$ F . Take
L = B[q and let F' be the image o f F by the isomorphism B E C ( L ) . Then [F')s= {f E L 139 E F', g 5 f} by Remark 1.6 and this &filter is not an m-filter. Now c) take an m-algebra L and a E L . Then [ a ) is an m-filter; besides, [a) is a &filter or not according as a E C ( L )or not. A filter which is both a &filter and an m-filter will be called an (m, O)-filter. The reader is urged t o transcribe the results o f this section to the case of the (m, d)-filters of an m-algebra. In particular the restriction of the functions in Theorem
1.7 yields an isomorphism between the complete lattice o f all (m, d)-filters of t he m-algebra
L and the m-filters of the Boolean m-algebra C(L).
1.21. Remark (lorgulescu [1984c]). Let us call m-congruence o f an LM-algebra
L , any congruence relation p on
L satisfying the stronger conditions (i) for every
A
h€H
L , c a r d H 5 m, if a h p b h (Vh E provided both meets exist, and
( a h ) h € ~ (, b h ) h E ~C
ahp
A
bh
H ) then
h€H
(ii) the dual condition. Suppose L is an m-algebra (cf. Definition 4.6.1) and let F be a &filter of
L. Then using Proposition 1.2.27 and F = i it is easily seen that m o d F is an m-congruence if and only if F is an (m,d)-filter. Let us call (m,d)congruence every 8-congruence which is also an m-congruence. Thus m o d F is an (m, 29)-congruence for every (m, 9)-filter
F
and as a matter of fact the
isomorphism constructed in Theorem 1.13 can be restricted to an isomorphism between the complete lattice of all (m, 29)-filters and the complete lattice of all (m,d)-congruences.
The reader is urged to transcribe other
results of this section t o the case of (m, 8)-congruences. Now we turn to further consequences of the previous basic results. Proposition 1.14 has a kind of converse for 3-valued Moisil algebras. Let
L
be a double pseudocomplemented lattice; by a ++-closed filter is meant
257
8-filters, 8-ideals and 29-congruences a filter
F of L such that x E F
+ x++
++-closed filter for every congruence p of
E F. It is obvious that ker p is a
L.
1.22. Theorem (Varlet [1972]).
Let L be a double pseudocomplemented lattice. T h e m a p ker is a bijection between O(L)and the set of ++-closed filters of L if and only i f L E M3. Sketch of proof. Sufficiency. If L E M 3 then i t s congruences coincide with
the congruences of the underlying double pseudocomplemented lattice, while the 3-filters are exactly the ++-closed filters. Necessity. In view of Theorem 4.4.4 it suffices t o show that
-
L is a dou-
ble Stone algebra verifying the determination principle. The latter property follows from the fact that the congruence
defined by (z
N
y
+ x* = y*
and z+ = y+) satisfies ker N= (1) = ker
1~ hence -= 1 ~ Further . one proves that if x+ A x++ > 0 then the ++-closed filter [zt)is distinct from ker p for every p E O ( L ) . This contradiction shows that zt A z++ = 0 and 0 one proves similarly that z* V x** = 1. 1.23. Proposition.
If L E LMn t h e n O(L)E O ( C ( L ) ) hence O ( L ) is a Heyting dgebra. Proof. Follows from Theorem 1.7, Proposition 1.14, Corollary 1.15 and e.g.
the dual of Example IX.3.5 in Balbes and Dwinger [1974].
0
The next two propositions were first proved by Beazer [1976] for 3-valued Moisil algebras, using the theory of double palgebras. 1.24. Proposition.
T h e following conditions are equivalent for L E LMn:
(i)
O ( L ) i s a Boolean algebra;
(ii)
C(L)i s finite;
(iii) L i s finite.
Filters, ideals and d-congruences
258 Proof.
(i) ($ O ( C ( L ) ) E B H (ii): By Proposition 1.23 and Berman [1973]. (ii) + (iii): Because L' = (C(L))["-'] is a finite Post algebra and PL : L --f L' is a monomorphism. (iii) + (ii): Trivial. 0 1.25.
ProDosit ion.
T h e following conditions are equivalent foT L E LMn:
(i) @ ( L )is a Stone algebra; (ii) C(L)i s a complete Boolean algebra. Proof. (i) H
O ( C ( L ) ) is a Stone algebra
H (ii) by Proposition 1.22 and
Beazer [1973].
0
1.26. Definition.
A class K: of 7-algebras is said to have the congruence extension property if for every A E K , every subalgebra B of A and every congruence p of B , there is a congruence p' of A such t h a t p'
n B2 = p.
1.27. Proposition (Boicescu (19841).
T h e class LMn has the congruence extension property. Proof. Let A E LMn and B , p as in Definition 1.26. The filter F generated by ker p in A is obviously a d-filter and we shall prove that m o d F n B 2 = p. If ( z , y ) E m o d F f l B2 then x , y E B and there is u E A such that
x A u = y A u and a 5 u for some a E ker p , therefore x A a = y A a, hence ( 2 , ~ )E mod o ker p = p by Proposition 1.14. Thus m o d F n B 2 E p, while 0 the converse is obvious from p = mod o ker p. 1.28. Remark.
It is known (see e.g. Pierce [1968], Lemma 3.1.8) t h a t if A is a 7-algebra and p,u E @ ( A )then p o u E @ ( A ) if and only if p o u = a o p, in which case p o u = p V u in the lattice @ ( A ) .The algebra A is said t o have
259
29-filters,29-ideals and 10-congruences
commuting congruences provided p o a = a o p for every p , a E @ ( A ) . 1.29. Theorem (Boicescu [1984]; cf. Beazer [1972] for M3).
Every L E LMn has commuting congruences. Proof. Let p l , p 2 E @(L).Then p h = modFh ( h = 1,2), where Fl and F .
L. Take (x,y ) E p1 o p2. Then there exist z E L and uh E Fh 1,2) such t h a t x A u1 = z A u1 and z A ug = y A u 2 . This implies
are n-filters of
(h =
x p l x A u 1 , yp2 yAu2 and x A u l A u 2 = y A u 1 A u 2 , therefore x A u l p 2 y A u l
x A uzp1 y A u2. Setting w = (x A u2) V ( y A u l ) and using t h e above
and
relations it follows that
i.e. x p ~ ~ a n d s i m i l a r l y y ptherefore(x,y) ~v, E p20p1. Thusplop,
and
p2
o p1
C p1
p20p1
o p2 by symmetry.
1.30. Remark. It is known that for every filter F of a Boolean algebra B and every x,y E B ,
(1.11)
x mod F y
* (x V jj) A ( 5 V y) E F
(x V jj) A (3 V y ) E F then x A u = x A y = y A u and conversely, if x A u = y A u for some u E F then using (2.18") we get because if u =
x V y V G = ( x A u ) V Y V G = ( y A u ) V j j V i i = 1, hence u (x
v
5 x
V y by (2.19") and similarly u
5
5 V y , therefore u
E F . This property was generalized by Varlet [1972] matter of fact it holds for LMn.
y) A (2 V y)
t o LM3 and as a
1.31. Proposition.
Let L E LMn (1.12)
OT
5
L E Mn. FOTevery F E Fln(L):
zm o d F y
*
n-1
A (pix V 9 ; ~A )
i=l
V piy) E
F
Filters, ideals and 19-congruences
260
and every congruence p i s obtained in this way: p = mod o ker p. Proof. Using Proposition 1.14 and Remark 1.12 we get
zmodFy @
* z modFy * (i = 1,...,n - 1) ,
cpizmodFcp;y
whence (1.12) follows by Remark 1.20. The equality p = mod o ker p was obtained in the proof of Proposition 1.14.
1.32. Definition. Let us say that: a) An element
z of a .r-algebra A has property R ("regular") provided the
map @ ( A )+ ?(A) defined by p b) An algebra A has property
H
[z],, is injective;
R whenever all of its elements have property
R; c) A class
K: of valgebras has property R if all of its algebras have property
R. Varlet [1972] proved t h a t a double pseudocomplemented lattice has pro-
R if and only if z* = y* & z+ = y + + z = y . In particular L M 3 has property R and this characterizes L M 3 within t h e class of double Stone
perty
algebras. One can prove more: 1.33. Proposition.
The class LMn has property R. Proof. Notice first that the element 1has property R, because if [l]= ,, [1lPz then ker
p1
= ker
p2
whence Proposition 1.30 implies
p1
= p2.
Then take z E L E LMn and p I , p 2 E O ( L ) such that [z],,
= [z],,.
1 (cf. Theorem 4.4.16) one sees Using the LMn-structures of [0, z]and [z, 1 that
pi
= P k fl [ O , Z ] ~
P k fl [z, 11'
(k = 1,2)
(k
= 1,2) are congruences on [O,z] and pk" =
are congruences on [z,11. Furthermore
261
&filters, 8-ideals and 8-congruences
and since x is the unit o f [0, x] it follows that pi = pk and similarly ply’ = p2”.
Now take (u, v) E p1. Then (u A z, v A X ) E pi = p; and ( u V x, v V z) E
p2”, therefore u = u V ( u A X ) pz u V (v A z) = ( u V
proving that ( u , v ) E p2. Thus
p1
v) A
(u V x) p2
C pz and similarly pz
0
pl.
1.34. Remark.
The concept of filter in a Boolean algebra has been given an equivalent definition, very appropriate t o the purposes of symbolic logic: a filter is the same as a deductive system. The latter term designates a subset
D
of the
Boolean algebra B such that
(i) 1E D and (ii) if z E
D
and z + y E
D
then y E
D.
B . Then 1 E F and from x E F and x -+ y =? vyi E F we obtain z ~ = yx A ( % V Y ) E F hence y E F . Conversely, suppose D is a deductive system. Then D # 0 because 1E D. If x E D and x 5 y then x ---f y = 1E D therefore y E D.
To prove the equivalence suppose first that
F is a filter
of
If x , y E D then from
x
-+
x A y = Z V ( x A y ) = f V y L. y
we obtain z + x A y E D by the previous property, therefore x A y E
D.
It turns out that the above properties can be extended t o LM-algebras. The point is a convenient generalization of the Boolean implication, invented by A. Monteiro in 1963 for 3-valued Moisil algebras and called weak implication; cf. Cignoli [1969a]. Then Cignoli [1969a], [1970] defined and studied the weak implication and the deductive systems of Moisil n- algebras. Thus the results presented below are due t o A. Monteiro and Cignoli.
Filters, ideals and d-congruences
262
-
1.35. Definition.
-
The weak implication on an LM-algebra is the operation
x
M
defined
y='PoxVy(Vx,yEL).
M
- -
1.36. Proposition.
The weak implication
has the following properties:
M
(1.13)
x
(1.14)
x
(1.15)
x
(1.16)
x
(1.17)
xVy
(1.18)
(x
(1.19)
1
(1.20)
z 5 y implies x -+
(1.21)
x 5 y iff (pix
(1.22)
2
(y
M
z)=l;
M
+ M
M
M
(y
M
y A z = (x
- -
M
Z = ( X
y)
M
Y)+
+ M
+ z ; M
z)A(y
+ M
2 = 2 ;
x=x;
--t
M
QoX=1.
Proof. Straightforward .
y =1 ;
(piy= 1
z);
M
M
M
--t
M
M
M
-
- M
(y + z ) = z A y
--t
M
; y) + (x
z)=(x
+
, Vi E I ;
z);
+ M
z);
by
263
6-filters, 6-ideals and 6-congruences
1.37. Definition. If X is a non empty subset of L then we say that x E L is a consequence of m
X if there is a finite subset { x l , ...,xm} of X such that A
j=1
xj + M
x = 1.
Let us denote by C s ( X ) th e set of all consequences of X ; we put also CS(0) = (1).
1.38. Proposition.
C s ( X ) = [ X ) , for every subset X & L E LMn. Proof. Cs(0) = (1) = [0)8. Then
m
U
~
A
O j=1
X j S Y
and the proof is completed by Remark 1.6.
1.39. Proposition. Let L E LMn. A subset S
0
L is a deductive s y s t e m if and only if it is
a 6-filter.
-
Proof. Let S be a deductive system. Then S m
x E Cs(S) then
A
j=1
xj
M
# 0
because 1 E S. If
x = 1 for some x l , ..., xm E S, therefore
Filters, ideals and &congruences
264
...
z1 + (zZ + M
M
-+ M
...) = 1 by (1.16),
-+ z)
(I,
M
hence after
c
m steps we obtain z E S. Thus Cs(S) C S and since S [S)e = Cs(S) by Proposition 1.38, it follows t h a t S = Cs(S) = [S)s is a &filter. Conversely, let S be a d-filter. Then 1 E S. If z E S and z then
(POX
-
M
yES
E S n C ( L ) = S' and
(Pox
(POY
M
-
= pox V(P0y = (Po(p0z v y ) = = (Po(z
M
E S' ;
y)
9 is obviously a filter of th e Boolean algebra C ( L ) ,hence a deductive system of C ( L ) by Remark 1.34, therefore 'pay E S c S, whence finally y E s. 0 but
1.40. Proposition (the Deduction Theorem).
Let L E LMn, X (1.23)
L
and z, y E
L. Then
y E C s ( X U {z}) H z + y E C s ( X ) . M
Proof. We use Proposition 1.38, Remark 1.6 and the obvious property you
wH
(POU
I (POW.
F
Let
show that F = { y E
= {y E L I z
--f
L 1 cp0(zl A ... A z,
E C s ( X ) } . The equivalences
5 'pay}, whence it is straightX U {z} F and F E F' for any F = [XU (5))s. The above proof i s
F is a $filter, &filter F' such that X F'. Thus also valid for X = 8 if we replace z1 A
forward t o show t h a t
y
M
5
A z)
... A 2,
by 1.
265
Prime filters $2. Prime filters
The aim of this section is the study of prime filters and prime d-filters in a Lukasiewicz-Moisil algebra. A special attention is paid to minimal and maximal prime filters. 2.1. Definition. Let L f LMd. For
L and G
F
(1.1)
F* = F n C ( L ) ,
(2.1)
G;= cp;’(G)
C ( L )we set
(Vi E I ) .
A proper (prime) d-filter is defined as a d-filter which is also a proper (prime) filter. We denote by PFld(L) the set of all prime 8-filters on L. By a masimald-filter is meant a maximal element in the family of proper 8-filters ordered by set inclusion. 2.2. Lemma (Cignoli [1969a]).
Let L E LMd, F a filter on L and G a filter on C(L).Then for every i ,j E I : (i)
G;is a filter on L ;
(ii) i (iii)
+ Gi
Gj;
(G;)*= G;
(iv) F* is a filter on C ( L ) and (F”)o
zF c (F*)l;
(v)
G is prime, or equivalently maximal, in C ( L )($ Gi is prime in L ;
(vi)
if F is prime in L then F* is prime, or equivalently maximal, in C(L).
Proof. Straightforward.
266
Filters, ideals and 29-congruences
2.3. Theorem (Cignoli [1969a], Georgescu and Vraciu [1969a], Georgescu
[197I d ]). Let L E LMd and F a 19-filter o n L. T h e n the following conditions are equivalent:
F is a m a z i m a l O-filter;
(i)
(ii) F* is a n ultrafilter o n C ( L ) ; (iii) F* is a p r i m e filter o n C ( L ) ; (iv) F is a prime 8-filter; (v)
either
(vi)
F is
'piz
E
F
OT t&a:
E F (Vi E I) (Vz E L ) ;
a m i n i m a l p r i m e filter.
Proof.
(i) H (ii): By Theorem 1.7. (ii) e (iii) e (v): By Proposition 1.3.18. (iii) e (iv): Because z V y E F e 'pox V 'poy E F*. (iv) (vi): Take a prime filter F' of L such t h a t F' c F . Then F'* c F* and F'* and F* are prime filters of C ( L ) by Lemma 2.2, hence they are maximal by Proposition 1.3.18, therefore F" = F*. Now it follows by Proposition 1.4 and Lemma 2.2 that F = (F*)o= (F'*), c F', therefore F = F'. (vi)
j
(iv): Trivial.
0
2.4. Remarks.
F is necessarily a &filter, because Lemma 2.2 implies in turn that (F*)o& F , F* is prime in C ( L ) , (F*)o is prime in L , therefore F = (F*)ois a &filter by Theorem 1.7.
a) Any minimal prime filter
b) Prime (or, equivalently, maximal) &filters coincide with minimal prime filters, by Theorem 2.3 and t h e above remark a). c) The map * establishes a bijection between PF16(L) and PFl(C(L)), by
Theorems 1.7 and 2.3.
Prime filters
267
Several characterizations of those lattices
L E DO1 in which maximal
Stone filters coincide with minimal prime filters were obtained by A. Monteiro [1954] and then by Cignoli [1971b], who started from a result of Cherciu
[1971]. 2.5. Proposition.
Suppose L E LMd, F as a 29-filter, 0 # S C L , F n S = 0 and x,y E S + x V y E S. Then there is a prime 19-filter P such that
PnS=Q)andFGP.
F* n (cp0S]c(~) = 0, otherwise x 5 cpos for some x E F* and s E S, which would imply cpos E F , then s E F , a contradiction. According t o Q and Theorem 1.3.10 there is a prime filter Q of C ( L ) such that F* 0 Q n (cpOS]c(~)= 0. Then P = Qo has the desired properties. Proof.
2.6. Remark. Proposition 2.5 yields corollaries similar t o Corollaries 1.3.11-1.3.14 of Theorem 1.3.10. Thus e.g. a) in a d-algebra every proper d-filter is included in a prime &filter and is
an intersection of prime &filters. Also,
b) given z,y E L with x
#
y, a necessary and sufficient condition for the existence of a prime d-filter F such that x E F and y $ F is cpox $ y.
2.7. Proposition.
The intersection of all maximal 9-filters of a n LM19-algebra L is (1). Proof. It follows from Theorem 2.3 that an element x belongs t o all maximal 6-filters
F of L if and only if cpOx belongs to all maximal filters F* of C(L),
and this is equivalent t o pox = 1 by Theorem 1.3.28, therefore t o x = 1 by (3.1.27) and (3.1.2). 2.8. Definition. Let us say t h a t a congruence (a d-congruence) is:
(i) proper if it is distinct from the universal congruence L2,and
0
268
Filters, ideals and 6-congruences
(ii) maximal if it is a maximal element in t h e family of all proper congruences (proper 2')-congruences)ordered by set-inclusion
C.
According t o a general definition of universal algebra, an algebra A i s called semisimple if the intersection of all its maximal congruences is the trivial
congruence A.A. 2.9. Corollarv.
The i n t e r s e c t i o n of a l l maximal 19-congruences of an LMb-algebra L i s
AL. Proof. From Proposition 2.7 and Theorem 1.13.
0
2.10. Corollary. LMn-algebras and Mn-algebras are semisimple. Proof. From Corollary 2.9 and Proposition 1.14.
0
2.11. Remark.
LM6 (LMN1S)-algebras are also semisimple, because maximal 6-congruences are identical with maximal congruences. 2.12. Proposition.
The following conditions are equivalent f o r a prime ideal I of an algebra
L E LM6: (i)
I
(ii)
In Ds(L) = 0;
i s a 2')-ideal;
(iii) for every x E L, e i t h e r x E I o r x* E I. Proof.
(iii): If x E I then 'plx E I hence x* = Cplz # I because I is proper. If x # I then 'plx# I hence the prime ideal I f l C ( L ) of C ( L ) (cf. the dual of Lemma 2.2) contains Cplx (6. t h e dual of Proposition 1.3.18).
(i)
j
(iii)
+ (ii):
contradiction.
If x E I n D s ( L ) then x E I , z* = 0 and x*
# I,
a
Prime filters
269
l(i) =+ l(ii): If z E I and cp1z (i!I then as for (i) + (iii) we get Cplz E I n C ( L ) , hence z V Cplz E I and since cpl(z V plz) = 1 it follows that z V qlzE I n Ds(L).
0
2.13. Remark. In this book there are several uses of t h e symbol Ker which should not be confused, even if they are not distinguished by subscripts. The first meaning is the one used in universal algebra: if A, A' are r-algebras then
(2.2)
kerA,At
: Hom(A,A') + @(A)
is defined as in (1.5.6), i.e. for every f E Hom(A,A'), (2.3)
*
~(ker~ f , )~ ~ ) f(z) = f
( ~ )( V ~ , YE A ) .
The first occurrence of this concept was in formula (1.3.8), in t h e particular case of lattices. As a matter of fact we are interested in t h e more particular case of algebras L , (2.4)
kerLp
L' from LM9 or LMNd. Note that in this case
: Hom(L,
L')
-+
Oo(L)
because if p;z(kerL,Lt f)p,y (Vi E I) then cpif(4
= f ( c p i 4 = f(cpiY) = Vif(Y)
(Vi E I)
therefore f(z)= f(y) i.e. z(kerL,Lt f)y. The next meaning of ker was first used in Theorem 1.13, again for L , L' E LM19 or LMNB: (1.4)
kerL : Oo(L)-+ F16(L)
,
(1.7)
ker p = { z E L I z p l }
(Vp E O O ( L ) )
A third and similar meaning will be introduced now. 2.14. Theorem (Cignoli [1969a], Georgescu [1971d]).
Let C = LM19 or C = LMNB. For every L E C there is a bijection between PFld(L) and Homc(L, L;'). Proof. Set Homc(L,L, UI) = H and KerL = kerL o k e r L , y i.e., for each
f EH,
Filters, ideals and d-congruences
270
K e u f = 1. E L I s(kerLLrJl ' 2 f)ll = f-1([1))
(2.5)
7
l(i) = 1 (Vi E I). As L;' is a chain by Lemma 4.6.11it follows that [l) is a prime d-filter
where 1 E Li',
of Li', hence it is straightforward t o check that f-'([l))is a prime 6-filter of L. Thus KerL :
H + PFlO(L).
Now we prove the existence of KerL', namely
(Keri'F)(x)(i) =
(2.6)
(
1
,
vix E F
0
7
vix $ F
(Vx E L) (Vi E I ) ,
for each F E PF16(L). Let f ( x ) ( i ) stand for the right side of
f(x) E ' iL
(Vx E L).
The next step is t o check that every
(2.6).Clearly
f
E
H . Using (3.1.11) we
see that for
x E L and i,j E I,
* Yjcpix E F H Fix E F =1* (f(x))) (.i> =1
f(vjx)(j) = 1 H
f(
(vi
e
7
proving that f(cpiz) = cpi ( f ( z ) ) .One verifies similarly, using Remark 1.3.5.
1.3.9,that f i s a bounded- lattice homomorphism. In view of Remark 3.1.29 (and of Remark 1.2.29 if C = LMNd) this shows that f E H . Then using Remark 1.2 we obtain a) and Proposition
x E ker f
* f(x) = 1 H f(z)(i) = 1
H cpix~F
satisfies
I) e
(V~EI)HZ€F,
f = F . Finally it remains ker g = F then g = f. This follows from
which proves t h a t ker
(Vi E
t o show that if g E H
271
Prime filters 2.15. Corollary (Georgescu [1971d]). If L E P19 t h e n a) every m o r p h i s m f :
b) L'\
L +L '\
is a surjection, and
has n o proper Post subalgebras.
Proof.
ra
(C(L)) , hence Proposition 4.1.6 yields an element x E L such that 'pix = u ( i ) (Vi E I). Notice that f = ker-' ker f by Theorem 2.14 and f ( u ( i ) ) = 1 or 0
a) Take a E
Lia. But
a can be viewed as a funtion a E
according as u ( i ) = 1or 0. Then f(x) = a because
f(z)(i) = 1 H 'pix f ker f H u(i) E ker f
* f (a(;)) = 1
($
a(;) = 1 ,
b) Apply a) t o the inclusion mapping of a subalgebra L of LiA.
0
2.16. Corollary.
T h e following conditions are equivalent
L E P19 and f E F119(L):
~ O T
(i) F is a maximal 19-filter;
(ii) L / F Proof.
(i) j ( i i ) : The morphism f = Ker-'F : L + is a surjection by Corollary 2.15, therefore L'\ FZ L / ker f by the homomorphism theorem 1.5.16. Thus it remains to prove that ker f = modF. First we apply (2.6) and obtain
Now if z ker f y then for each i E I, (2.7) and Theorem 2.3 imply that either ( p i x , (piy E F in which case we set uj = 'pixA (PigI or (pix, Cpiy E F in which case we set u;= pix A p;y. In both cases
Filters, ideals and 29-congruences
272
that is z m o d F y . Conversely, (2.8) implies that for each i E I,if via: E F then (piy A ui E F hence (pig E F and similarly (p;y E F + pix E F therefore z ker f y by (2.7).
(ii) + (i): Suppose F’ is a proper 29-filter of L such that F E F‘. Then = {2 I z E F’} is a proper &filter of L/F by Proposition 1.19. As 0 remarked after Definition 1.1, this implies F? = {i},that is F‘ = F .
2.17. Proposition. T h e following conditions are equivalent f o r L E LM29 and F E Fl(L):
(i) F is mazimal; (ii) there is a p r i m e filter G of C(L) (necessarily unique: G = F’) such that F = GI. Proof.
(i) + (ii): In view of Lemma 2.2 we obtain in turn that F* is prime in C ( L ) , F C (F*)1 and (F*)l is prime in L , hence proper; therefore F = ( F * ) l . If F = G1then F’ = (Gl)* = G by Lemma 2.2. (ii) =$ (i): If F’ is a proper filter of L such that F F’ then G = F’ (F’)’ C C(L) hence F’ = (F‘)’ therefore F = F’ by Lemma 2.2. 0 2.18. CorolI a ry . If L E LM29 and F E PFl(L) then F’ = (F*)oand F” = ( F * ) l are the unique m i n i m a l p r i m e filter F’ F and the unique m a x i m a l filter F” 2 F , respectively.
s
Proof. In view of Lemma 2.2, F’ is a maximal filter of C(L) and F‘ C F E F”. But F’ is a maximal &filter by Theorem 1.7, therefore a minimal prime filter by Theorem 2.3, while F” is a maximal filter by Proposition 2.17. If P is a maximal filter such that F P then F* & P” and F’ E F”’, therefore P” = F’ = F” hence z E P implies (pox E P’ = F”’ F” then z E F”, which shows that P E F”, therefore P = F”. The uniqueness of F’ is
s
273
Prime filters proved analogously.
A. Monteiro [1974]has shown that for L E D01, every prime filter is included in a unique maximal filter if and only if L is n o r m a l i.e. for every x,y E L satisfying x Ay = 0, there exist u , v E L such that X A V = y A u = 0 and u V v = 1. Then it follows from Corollary 2.18 t h a t every L E LM9 is normal; this can be also proved directly taking u = 'plyVx and v = cplxVy.
2.19. Definitio n. Let L E LM9. Denote by Rad(L) the intersection of all maximal filters of L. For every proper filter F of L let Rad(F) stand for t h e intersection of all maximal filters of L that include F . 2.20. Proposition. Let L E LM19 and set M = PFl(C(L)). Then
(2.9)
Rad(L) =
FEM
Fl = Ds(L) .
Proof. The first equality follows from Proposition
Rad(L) = 9';
( FEM n
F ) = cp~'([l))
2.17,implying further
= Ds(L) .
2.21. Proposition. Let L E LM9 and F a proper filter of L. Then Rad(F) = cpT'(F). Proof. Let
MI be the family
L which include F that M1 = {G1 I G E
of those maximal filters of
M z = { G E PFl(C(L)) I F* G G } ; we claim M z } . If F' E M1 then F' C F", while Proposition 2.17 shows that F' = ( F f * ) land F'* E PFl(C(L)), hence F'* E M2. Conversely, suppose F' = G1 where G f M 2 . Then F' is maximal by Proposition 2.17 and 2 E F implies cplx E F* G G hence x E GI = F', proving that F 5 F', therefore F' E M I . Now using Corollary 1.3.14in C(L) for F* we obtain and
Rad(F) =
n
GEM2
cp;l(G)=cp;l(
n
GEM2
G)
=
Filters, ideals and 19-congruences
274
The next result connects filters t o the operations Ch. 4, $3.
and
% studied in
2.22. Theorem. The following conditions are equivalent f o r L E LMn: (i)
L is a Boolean algebra;
(ii) Rad(L) = [l);
(iii) every proper filter F is a n intersection of maximal filters, i.e. Rad(F) = F ; (iv)
x
+ y = ( P ~ - v~ y;x
Proof.
(i) + (iii): By Corollary 1.3.14 and Proposition 1.3.18. (iii) + (ii): In particular Rad([l)) = [l); but clearly Rad([l)) = Rad(L ) . (ii) + (i): As x A x* = 0, it remains t o prove that x V x* = 1. But x V x* E D s ( L ) , because (x V x*)* = 0 and D s ( L ) = [l) by Proposition 2.20. (i) + (vi): By Proposition 4.3.6. (vi) + (v): By (4.3.25) and (3.1.27) where 0 is 1. (v)
+ (i): The
(1
+ x) V (x +- 0)
= 1, that is : ( p ; ~ V c p ~= - ~1,xi.e. ( P ~ - 5 ~ X(pix.
hypothesis yields
xVcp,-lx = 1, whence for each i c I Thus (pix = (P,-~X (Vi E I ) , which shows that L (i) + (iv): By Proposition 4.3.6.
= C ( L ) E B.
275
Prime filters (iv)
+ (i): x* V x = qn-1x V I = x =$ x = 1.
0
2.23. Remark. Properties (i)-(iii) in Theorem 2.22 are equivalent even for L E LM6 (same proof).
A. Monteiro (cf. A. Monteiro [1974]) raised the problem of whether a bounded distributive lattice in which every proper filter is an intersection of maximal filters and dually, is a Boolean algebra. The answer is negative, as was shown independently by Balbes [1972] and Adams [1974]. However for
L E LMn the answer is positive; cf. Theorem 2.22. 2.24. Proposition.
The fol2owing conditions are equivalent for L
E LMt9 and
F E PFl(L):
(i) F is maximal; (ii) D s ( L ) Proof.
c F.
*
(i) (ii): By Propositions 2.19 and 2.16. (ii) =$ (i): Let F' be a filter such that F c F'. Take 5 E F' - F. From yl(x V p1x) = 1 we infer I V Flx E D s ( L ) 5 F and since F is prime it 13 follows that $311 E F 5 F', therefore 0 = I h Cplx E F', i.e. F' = L.
In the case of LMn-algebras we can say much more about prime filters. 2.25. Theorem (Boicescu [1984]). T h e following conditions are equivalent f o r L E LMn and a proper filter F of L:
(i)
F is prime;
(ii) there is a prime filter
F' F ;
(iii) ihere is a unique minimal prime filter F'
F;
276
Fiiters, ideals and 6-congruences
(iv) F* is prime in C ( L ) ; (v)
there is a prime filter G o f C ( L ) (necessarily unique: G = F*) and i E (1, ...,n - 1) such that F = G;;
(vi) F is included in a unique maximalfilter. Proof.
(i) j (iii): By Corollary 2.17. (iii) + (ii). Trivial. (ii) +- (iv): F’* E PFl(C(L)) by Lemma 2.2 and F’* = F*.
F“
C_ F*, therefore
c
F (F*)n-l by Lemma 2.2. We prove that F = (F*)jfor j = max {il(F*)iC F } . If j = n-1 then (F*),-lC F hence F = (F*)n-l. If j < n - 1suppose by way of contradiction the existence of x E F - (F*)j. Then ‘pjx @ F’ hence cpp E F*. But (F*)j+l F hence there is y E (F*)j+l such that y @ F . Using Proposition 3.2.3we obtain y 2 x A Cpjx A ( ~ j + E ~ yF therefore y E F , a contradiction. If F = Gi for G E PFl(C(L)) then G = F’ by Lemma 2.2. (v) +- (i): By Lemma 2.2. (i) 3 (vi): By Corollary 2.17. (vi) + (v): F is an intersection of prime filters F’ by Corollary 1.3.14. Each F’ is included in a maximal filter P by Corollary 2.18. But F 5 P therefore P is unique by hypothesis. Then P* and each F” are maximal in C(L)by Lemma 2.2 and also F” P*,therefore F‘* = P* for each F‘. According t o (iv) 3 (v), for each F‘ there i s a j such that F‘ = ( P * ) j ,which implies F = ( P ” ) k ,where k is th e least j. This representation is unique with 0 respect t o P* again by Lemma 2.2. (iv) 3 (v):
(F*),
c
Comment . Theorem
2.25 shows that if 6 = n certain properties of &filters can be
(iii) extended t o filters. Thus the equivalence (i) ($ (iv) generalizes (iv) in Theorem 2.3; (i) ($ (iii) and (i) (vi) generalize Corollary 2.18 and the lattices sharing this property have been characterized by A. Monteiro
[1974]. In that paper it
is also proved that for
L
E
D01,every proper filter
277
Prime filters
which includes a prime filter is also prime if and only if L is fully normal; cf. Definition 4.3.14. The above equivalence (i) e (ii) implies that every LMn-algebra is fully normal; cf. Proposition 4.3.16. The equivalence (i) ($ (v) generalizes a result of Cignoli [1969a] on Moisil n-algebras (see also Cignoli [1975a]). 2.26. Corollary (Cignoli [1969a], [1975a]). If L E LMn then t h e poset PFl(L) is the cardinal s u m of a set of chains of cardinality 5 n - 1 each: (2.10)
{G; I i = 1, ...,n - 1)
, G E PFl(C(L)) .
Proof. Taking into account Theorem 2.25 it only remains to prove that every two elements belonging to distinct chains (2.10) are incomparable. If Gi C Gi for G,G‘E PFl(C(L)) then G = (G;)’ (G>)* = G‘, therefore
G = G‘.
0
2.27. Corollary (Epstein [1960], Traczyk [1963]). I f P E Pn then each chain (2.10) has exactly n - 1 elements. Proof. For each G E PFl(C(L)) and each i = 2, ...,n - 1, from (pic,-i = 1 E G and ( P ; - ~ C , - ~ = 0 @ G we infer c,-i E G; but c,-i @ Gi-1, therefore
Gi-1
c Gi.
0
The converse of the above result will be the object of the next section 2.28. Corollary. If L E LMn, F’,F” E PFl(L) and F’, F” are incomparable t h e n F’VF” = L. Proof. If F’V F” is a proper filter then there is F E PFl(L) such that F’VF” F , which implies F” C F* and F”’ E F’ hence F” = F” = F’”, therefore Theorem 2.25 yields F‘ = (F*)i and F” = (F*)j for some i , j . 0 This contradicts the incomparability of F’and F”.
278
Filters, ideds and d-congruences
2.29. Proposition. If L E LMn is infinite then card L
5 card PFln(L) = card PFl(C(L)) ,
Proof. It follows from Remark 2.4.c and Corollary 2.26 that (2.11)
cardPFl(C(L)) = cardPFln(L) 5 cardPFl(L)
5
5 ( n - 1)cardPFI(C(L)) . But C ( L )is infinite by Proposition 1.25, hence cardC(L) 5 cardPFl(C(L)) (Makinson [1969]), therefore PFl(C(L)) is infinite and (n-l)cardPFl(C(L))
= cardPFl(C(L)) which together with (2.11) yield
cardPFl(L) = cardPFl(C(L)), while cardL Makinson result.
5 cardPFl(L)
again by the 0
Proposition 2.29 cannot be generalized to infinite 6; thus e.g. Liq is infinite while cardPF1d(Lhq) = 1 as noticed after Remark 1.2. 2.30. Remark. The theory presented in this section can be extended in several ways. One of them has been already mentioned: the passage from LM-algebras to bounded distributive lattices. Another idea is followed in Sicoe (19711: the study of the ideals of the form c p i ( K ) or cp;l(IC), where i E I is fixed and I< runs over the ideals of L. lorgulescu [1984c] has obtained a natural generalization of Theorem 2.3 and several related results within the framework of m-concepts; cf. Remarks 1.3.31 and 1.20. In particular it is noted that every maximal 9-filter which is also an m-filter is obviously a maximal (m, d)-filter but it is not known whether these two concepts coincide and their relationship to the concept of maximal 19-filter is also an open problem. lorgulescu [1984c] also proved that in the d-valued algebras (d-infinite) the equivalence (i) % (v) in Theorem 2.25 does not hold.
The Cignoli problem
279
53. The Cignoli problem Epstein [1960] and Traczyk [1963] proved t h a t i n every Post algebra following property
L the
P holds: th e set PFl(L) o f prime filters of L is a cardinal
sum of ( n - 1)-element chains. In his thesis, starting f r o m Moisil algebras, Cignoli gave a new proof o f the above theorem (cf. Corollary 2.26) and asked whether the converse was true. More precisely: is every Moisil n-algebra w i t h
P a Post algebra? In this section we prove t h a t the answer is affirmative even in the more general case when L E LMn; cf. Boicescu [1984]. property
3.1. Lemma.
Let L E LMn, with property P . If F E PFl(C(L)) let Fl
c F2 c ... c
be the chain of prime filters in L determined by F (cf. L e m m a 2.2). FOT each i = 2, ...,n - 1, there exists x b E L, such that:
F,-1
(i) x & E ~ - - - * ( i = 2 ,...,n-I),
(iii) 'pix> =
= ... = ' p n - l x $ .
3.2. Lemma.
Let B be a n n-valued Post algebra and F E PFl(C(B)) and i E ( 2 , ...,n1). T h e n the elements
satisfy conditions (i)-(iii) of L e m m a 3.1. Conversely, every element 2; satisfying conditions (i)-(iii) of L e m m a 3.1 has a unique representation
Filters, ideals and &congruences
280 of the f o r m (3.1), n a m e l y yk = 'pix.
Proof. It follows from (3.1) that 'pjx; = 0 for all j < i and 'pix; = y$ for all j 2 i. Therefore (ii) and (iii) hold. In particular ' p i - 1 ~ ; = 0 hence x; $ F,-1, while 'pixi = yk E F hence 3% E F;. Thus (i) holds and the representation (3.1) is unique. Conversely, if Z> = cn-i
&EB
satisfies conditions (i)-(iii) of Lemma 3.1, then
A 'pix; foIIows from
3.3. Lemma. Under the hypotheses of L e m m a 3.2, there exists a finite subset 'Po C PFl(C(B)) such that
Proof. In view of Proposition 1.3.9, the elements of SpecC(B) are the sets L - F , F E PFl(C(B)). But y> g! L - F hence L - F E s(y$) (cf. Notation 2.1.5) for each F E PFl(C(B)), therefore SpecC(B) = U{s(yk) I F E PFl(C(B))}. Now the compactness of SpecC(B) and Corollary 2.1.6 imply the existence of a finite subset Po S PFl(C(B)) such that
whence (3.2) follows by injectivity of s.
0
3.4. Theorem. If L E LMn and satisfies condition P , t h e n L i s a n n-valued Post afgebra. Considering the canonical monomorphism (4.1.3) F'L : L + C(L)["-'I, it follows that L can be identified with the LMn-subalgebra PL(L) Proof.
The Cignoli problem
281
C(L)["-l]and it is sufficient to prove the theorem for PL(L). As C(L)["-'] is a Post algebra and therefore has property P , it follows that for any F E PFl(C(C(L)"+'])), the chain of prime filters l increasing. As in C(L)"+'], ( ~ ) l ~ i ~ nis- strictly
of the n-valued Post algebra
(3.3)
C(PL(L))= C(C(L)["-']),
P,it follows that the chain of prime filters in PL(L) F1 n PL(L) c FZ n PL(L) C ... C Fn-1 n PL(L).
and PL(L)has property
F
determined by
is
According t o Lemma 3.1, there exists z& E PL(L) satisfying conditions
(i)-(iii). However zk belongs t o the Post algebra C(L)"+'] and therefore by applying Lemma 3.2, it follows that there exists y b E F , such t h a t zb = c,-i A yk, for any i = 2, ...,n - 1 and any F E PFl(C(PL(L))). According t o Lemma 3.3 there exists a finite subset Po c PFl(C(PL(L)))
V
such t h a t
yk = 1. But in
C(L)fn-']
FEPo
and because
V
FEPo
z& E PL(L),it follows that c,-; E PL(L)i = 2, ...,n-1.
Using again (3.3) we see that PL(L) is an n-valued Post algebra.
0
Comment.
The above proof implies in fact (3.5)
4;n-i
= V{z$ I F E PFI(C(PL(L)))}
(i = 2, ...,n - 1)
2 z$ for all F and if z 2 zb (VF) then x 2 c,-i by (3.4). As a matter of fact (3.5) holds in PL(L)because cn-i E PL(L). Conversely, a direct proof of (3.5) i.e., for L and without using Lemma 3.2 and 3.3because c,-i
would yield a direct proof of Theorem 3.4-i.e.
without using Post algebras
or topological arguments. We are able t o give such a proof in the case when
L is complete.
282
Filters, ideals and 19-congruences
3.5. Proposition. Let L E LMn be complete and with property P. Then (3.6)
( i = 2, ...,n - 1) ,
V { x b I F E PFl(C(L))} = cn-;
where x f . are the elements constructed a'n Lemma 3.1 and cn-i satisfy (4.1.9)(cf. C o r o k l ~ y4.1.9). Proof. We adopt (3.6) as a notation and prove (4.1.9), i.e. cpjc,-i = 1or 0 according as j
Let i E (2,
2 i or j < i. It suffices t o prove ( P ; c , - ~= 1and ( ~ i - ~ c = ~ -0.i
...,n - 1).
F E PFl(C(L)), from cn-i 2 x ; and x ; E F; we obtain (Pic,-; 2 (Pix; E F , hence Propositions 2.7 and 1.3.18 imply For every
(Pic,-i
E
n {F I F E PFl(C(L))} = (1) .
Now suppose by way of contradiction that exists F E PFl(C(L)) such that follows t h a t
~ p i - ~ A ~ -cpix; i
we infer x;,
5 pix;
E
(Pi-1Cn-i
which contradicts (3.7).
#
(~i-~c,-i
0. Then there (pix& E F it
F and since
F ,hence
V x ; . Therefore cn-i
turn
E
5 q ; x & V 2% and this
implies in
The Cignoli problem
283
We reminded above t h a t if L E DO1 and L is an n-valued Post algebra, then L has property P . The converse is not true, as was proved by Chang and Horn [1961]. They have considered L = L:, where S = {sili E w @ 1). Obviously L E DO1 and let L1 be the subset of L formed by the elements such t h a t f ( s n ) = c2, for any n E w, n 2 n o , for some no E w and f(sw) = c1 or c2, or f ( s n ) = co, for any n E w, n 2 no, for some noE w and f(sw) = co. One proves that L1is a sublattice with 0 and 1of the lattice L and has property P , but L1 is not a three-valued Post algebra. One can say even more:
3.6. Proposition. L1 E DO1 defined above has no structure of three-valued LukasiewiczMoisil algebra. Proof. As
L1 satisfies condition P
but cannot be a three-valued Post alge-
bra, this follows from Theorem 3.4.
0
This proposition was initially proved by Cignoli [1969b]. Chang and Horn [1961] and then Traczyk [1963] gave necessary and sufficient conditions for a bounded distributive lattice having property P t o be a Post algebra, but in those characterizations appear the constants ci from the definition of a Post algebra.
3.7. Corollary. If L E DO1 and satisfies property P , then the following are equivalent:
(i) L is a n n-valued Post algebra;
(ii) L has a structure of n-valued Lukasiewicz-Moisil algebra. 3.8. Corollary. If L E DO1 and satisfies property P , then L has a unique structure of n-valued Lukasiewiez-Moisil algebra compatible with the lattice structure. Proof. By Theorem 3.4 it follows that L has a structure of n-valued Post algebra. This structure is unique (cf. Balbes and Dwinger [1974] p. 192
284
Filters, ideals and &congruences
or Exercise X.2.8) and it determines uniquely the structure of LukasiewiczMoisil algebra; cf.
(4.1.11)and Corollary 4.1.9.
0
3.9. Corollarv.
If L is a n n-valued P o s t algebra then L has a unique structure of n-valued Lukasiewicz-Moisil algebra compatible with t h e lattice structure of L.
3.10.Corollary. If L E DO1 and PFl(L) as a cardinal s u m of chains of cardinality 2, t h e n the following are equivalent:
L i s a double pseudocomplemented lattace;
(i)
(ii) L E M3; (iii) L is a three-valued Post algebra; (iv) L is a double Stone algebra; (v)
L i s a double pseudocomplemented lattice and D s ( L )n
# 0.
Proof.
(i)
by Varlet [1968]. (ii) + (iii): By Theorem 3.4. (iii) + (iv): By Theorem 4.4.4. (iv)
(iii)
=$-
(ii): Proved
+ (i): Obvious. + (v): There exists c1 E L , such that (plcl= 0 and cpzcl = 1. By
Proposition (v)
4.4.6,c1 E D s ( L ) n
(i): Obvious.
m.
0