Chapter 1 Lattices, Universal Algebra and Categories

1

CHAPTER 1 LATTICES, UNIVERSAL ALGEBRA AND CATEGORIES

The first t w o chapters of this book establish all prerequisites necessary

to t h e study of Lukasiewicz-Moisil algebras. Most important is the latticetheoretical background, presented w i t h full proofs: $1deals w i t h lattices i n general, $$2,4 are devoted to the classes of lattices relevant t o our theme, while the theory of filters, ideals and congruences and t h e representation theory i n terms o f topological dualities are treated in $3 and in Chapter 2, respectively. T h e other tools, provided by universal algebra and category theory, are dealt w i t h in t h e last t w o sections of this chapter.

$1. Posets and lattices The main topics treated in this section are: lattices as partially ordered sets and as algebraic systems, morphisms, sublattices, closure operators, the

MacN eille corn pletion. 1.1. Definition. A p a r t i d y ordered set or poset is a couple (f',

5 ) where P is a non-empty set and 5 a relation of partial order on P, i.e. 5 is reflezive (x 5 x) antisymmetric (x 5 y and y 5 x +- x = y) and transitive (x 5 y and y 5 2 * x 5 2). T h e relation

z

<

o f strict partial order associated to

< y -e(x 5 y and x # y).

Then

2

is defined by

< is transitive and fulfils x < y + y f

f x for every x ; moreover, x 5 y Thus in a poset we can use both relations 5 and

which implies

5

tj

<

(x <

y or

2,

x = y).

as well as t h e dual

relations 2 and >, defined by x 1 y H y 5 x and x > y (j y < x, respectively. The duality principle for posets enables one to duplicate

2

Lattices, universal algebra and categories

theorems by interchanging

that

2

5

with

is also a partial order and

2

and

> a strict

< with >; it is based on the fact partial order. We say t h a t

( P ,2)

is the poset dual t o ( P ,I). In the sequel we shall denote various partial orders by the same symbol < whenever

this can be done without risk of confusion. A similar convention

will apply t o the operations and constants dealt with in this book.

1.2. Examples.

The direct product of a family of posets (Pi,
(

II P;, I),

iEI

I is defined componentwise, i.e.

If t h e index set I is also endowed with a total order (cf. Definition 1.5), then the lezieographic product o f the above family is the poset ( II P , , < D ) , i€I

where

If, moreover, PinPj = 8 whenever i family is the poset

(1.3)

(

# j,then t h e ordinal s u m of t h e above

U P,, I),where

ieI

xIy@(3ix

5i

y ) or ( 3 i 3 j i < j and x ~ P and i YEP,)

1.3. Example.

If ( P ,5 ) is a poset and X an arbitrary set, the set Px = {f I f : X -+P } is endowed with the partial order defined pointwi3e, i.e.

1.4. Definition.

Let (P ,<) be a poset and S 2 P . By a lower bound (an upper bound) of S is meant any x E P such t h a t x 5 s (s 5 x) for every s E S. If

3

Posets and lattices

x of S belongs to S then z is said to be the least or f i r s t (greatest or l a s t ) element of S . If t h e set P itself has a least (greatest) element, this is usually denoted by 0 (by 1) and called the zero ( o n e ) of P . A poset having both 0 and 1 is said to be bounded. The infimum ( s u p r e m u m ) of S is its greatest lower bound (least upper bound): g.1.b. S or inf S or A S or A z (1.u.b.S or s u p s or V S or V x) for

the lower bound (upper bound)

X€S

XES

short. A m i n i m a l ( m a x i m a l ) element o f

S is an element rn E S such that there is no element s E S satisfying s < m (rn < s ) or, equivalently, such that for every s E S : s 5 rn + s = rn (rn 5 s + s = rn). For a given subset S o f

P some or all of the above elements may not

exist. O n t h e other hand t h e infimum (supremum) is unique whenever it exists. Moreover, if the least (greatest) element o f S exists, it is unique and is also the infimum (supremum) of S and t h e unique minimal (maximal) element o f S.

1.5. Definition.

x 5 y or y 5 x (or, equivalently, if 5 < y or y < 2 or z = y); otherwise we say t h a t x and y are incomparable and write x # y. A totally ordered s e t or t o s e t or c h a i n

T w o elements z, y of a poset are said t o be comparable if

is a poset in which every two elements are comparable.

1.6. Examples. T h e lexicographic product and t h e ordinal sum of a family o f tosets is a toset

(cf. Example 1.2).

1.7. Definition. Suppose the elements z, y of a toset fulfill z


and there is n o element z

x < z < y . Then we say t h a t y is t h e successor of x and write or, equivalently, t h a t x is the predecessor of y and write x = y - .

such that

y = I+

The successor (predecessor) o f an element is unique provided it exists,

Lattices, universal algebra and categories

4

because if both y and y‘ are successors of

x then e.g. x < y 5 y’ which

is

possible only for y = y’. Also,

(1.4)

x 5 y H x- < y H x < yt

x 5 y and x- # y would imply x 5 y 5 x-, while would imply x- < y < 2. for

5-


and

x $y

1.8. Definition.

A m e e t semizattice (join semilattice) is a poset ( L , 5 ) such that every two-element subset { x , y } has an infimum (a supremum); this element is usually denoted by

x A y (by x V y) and called t h e m e e t ( j o i n ) of x and

y. A Zattice is a poset which is both a meet semilattice and a join semilattice. The following identities are valid in a meet semilattice and in a join semilattice, respectively.

I

(1.5’) (1.5”)

xAy=yAx x V y =y V x

(1.6’) (1.6’)

(x A y) A z = x A (y A z ) (x v y) v z = x v ( y v z )

(1.7’) (1.7”)

xA x =x

x V x =x

and in a lattice

(1.8’) (1.8’)

I

(commutativity);

I

(associativity);

(idempotency );

x A (x V y) = x x V (x A y) = x

(absorption).

Conversely, suppose A, V are binary operations on t h e non-empty set

L. If A fulfils (1.5’)-(1.7’) then L can be made into def ining

(1.9’)

x5y

H x A y =x

and if V fulfils (1.5”)-(1.7”) definition

a meet semilattice by

then

L becomes a join semilattice via the

Posets and lattices (1.9”)

5

5

< y *x v y =y

In other words, meet semilattices and j o i n semilattices can also be defined as algebraic systems satisfying axioms (1.5’)-(1.7’) and (1.5”)-(1.7”), respectively, while lattices can be defined as algebraic systems ( L ,A, V) satisfying (1.5)-(1.8) (as a matter of fact idernpotency follows from absorption). T he lattice ( L , V , A ) is called the dual o f (L,A,V). T he following properties are useful and easily proved:

(1.10’)

z < x A y ~ z < xand z < y ,

(1.10”)

x V y < z ( ~ x < z and y < z ,

(1.11’)

z < yJxA z< y A z,

(1.U’)

x
(1.12’)

x < y and t < v + x A t < y A v ,

(1.12)

x s y and t < v + x V t < y V u ,

(1.13)

~ = y H x A y = x V y .

T he elements 0 and 1, provided they exist, fulfil

(1.14’)

x A 0 =0 ,

(1.14“)

2

(1.15”)

v 1= 1 , x A 1= z , x V 0 =x ,

(1.16’)

xAy =lH z=y =l,

(1.16’)

xvy=o*x=y=o,

(1.15’)

and each of t h e above properties (1.14’)-(1.16”) is characteristic for the corresponding distinguished element. Thus e.g. if L is a lattice and

a

*

2

xAy =

= y = a then since a A (z V a) = a by (1.8’) it follows t h a t x V a = a

for every x, i.e. a = 1.

According to a general usage, posets, semilattices and lattices will sometimes be designated only by their underlying sets.

6

Lattices, universal algebra and categories

1.9. Examples. Suppose Li, i E I , are lattices (meet semilattices, join semilattices); then so are their direct product and their ordinal sum. For the direct product the operations are defined componentwise, i.e.

(1.17')

( 2 i ) i c l A (9i)iG.I = (zi

(1.17")

(2i)icl V ( y i ) i c l

Ai

Yi)icl

7

= (xi V i Yi)icl

7

while for the ordinal sum t h e meet (join) is defined by (1.9') (by (1.9')) if z

E L; and y E Lj where i

# j , and coincides w i t h

the meet (join) in Li if

both z,y E L; (cf. Example 1.2).

1.10. Example.

If L is a lattice (meet semilattice, join semilattice), so is t h e poset Lx constructed in Example 1.3. Here the meet and j o i n are defined pointwise, 1.e.

Lattices can be also characterized as posets in which every finite nonempty subset has an infimum and a supremum. Therefore the following concept is stronger than t h a t o f a lattice.

1.11. Definition. Let rn be an infinite cardinal number. A n rn-complete lattice is a poset in which every subset o f cardinality at most rn has an infimum (also called

m e e t ) and a supremum (also called join); cf. Definition 1.4. B y a complete lattice is meant a lattice which is rn-complete for every rn, or equivalently, a poset in which every subset has a meet and a join.

1.12. Remark. In particular A 0 = 1 and V 0 = 0. It is useful to note t h e somewhat surprising fact t h a t the existence of meets for arbitrary subsets, or equivalently the existence of joins for arbitrary subsets, is a sufficient condition for a poset to

7

Posets and lattices

be a complete lattice. The reason is t h a t the 1.u.b. (g.1.b.) o f a set S is the

g.1.b. (1.u.b.) of the set o f all upper bounds (lower bounds) of S. From a practical point of view we check separately t h e existence of 1and of meets for arbitrary non-empty sets, or the dual conditions.

1.13. Examples. Every finite lattice and t h e lattice

( P ( X ) ,c) of

all subsets of a set

X are

typical examples of complete lattices; in t h e latter lattice meets and joins coincide w i t h set-t heoretical intersections and unions, respectively. Si milarly, the family of all countable subsets of an infinite (uncountable) set is an Nocomplete lattice (which is not complete).

1.14. Definition.

A closure operator o n a poset P is a mapping : P t P such t h a t x 5 x', ''2 = x' and x 5 y + x c 5 y'. A Moore f a m i l y o f a poset P is a non-empty subset M C_ P such t h a t for every x E P the set {y E M 1x 5 y} has a least element.

1.15. Remark. There is a bijection between t h e closure operators and t h e Moore families of

P: it maps a closure operator to t h e Moore family { z E P I x = xc} = {x' I x E P } and conversely, a Moore family M is sent to the closure operator ' defined by xc = least element of t h e set {y E M I x 5 y}. See

a poset

e.g. Balbes and Dwinger [1974], Theorem 2.4.11.

1.16. Remark. In view of Definition 1.14 and Remark 1.12, a Moore family plete lattice

L

is itself a complete lattice

the order inherited from

(1.19')

M of a com-

( M ,infM, supM) w i t h respect t o

L:

infMX = inf X

,

(1.19') supMX = inf {y E M I x 5 y (Vx E X)} , for every X

c M . A n alternative characterization makes use of t h e operator

Lattices, universal algebra and categories

8 from Remark 1.15: SUPMX = (SUPX)"

(1.20)

1.17. Examr.de.

X c P define X + = { y E L I x 5 y V x E X} 5 x V x E X}. Then

Let P be a poset. For every

I

and X - = {y E L y

(ii) X 5 Y

+ Yf

(iii) the maps

+-

s X + & Y-

and -

X-;

are closure operators.

Property (i) is easily checked. From X+ therefore X

Y-

cY

c X + and (i) we get X C X + - ,

implies X C_ Y+- hence Y+ C X + by (i) and similarly

X-. Now (ii) implies th a t X C_ Y

c

+ X + - c Yt-.

Finally f r o m

X+X + - we obtain X+ C X t - + by (i),hence X t - + - C - X+- by (ii), therefore X + - + - = X + - . Similar proof for t h e second half o f (iii). 1.18. Definition. Let P and

P'

be posets. A mapping f

:

P + P' is said to be iso-

t o n e or increasing ( a n t i t o n e or decreasing) if z 5 y f(x) 5 f(y) (if x 5 y + f ( y ) 5 f(x)). T h e mapping f is called an i s o m o r p h i s m (a dual i s o m o r p h i s m ) provided it is a bijection and b o t h f and titone). In this case the posets write P

f-'

are isotone (an-

P and P' are said to be isomorphic and we

P'.

1.19. Comment. If f is an isomorphism so is

f-l;

as a matter o f fact the relation of being

isomorphic is an equivalence relation in th e class of all posets. Since isomorphic posets can be viewed as being identical, it is a natural idea to associate with each class of isomorphic posets a symbol called the order t y p e of all the posets in th a t class. As a matter of fact a theory has been developed which associates order types to classes of isomorphic tosets. In this theory

Posets m d lattices each n E

lN

9

is chosen as the order type of the set (1, ...,n} endowed with

the usual order. Then the class of all order types is equipped with an order relation which extends the order of lN.

1.20. Definition. Let L and L’ be meet (join) semilattices. A mapping f : L -+ L‘ is said to be a meet (join) homomorphism if f ( x A y) = f(x ) A f ( y ) (if f(x V y) = f(x) V f(y)). Meet homomorphisms and join homomorphisms are also designated under the common name of semilattice homomorphisms. If L and L’ are lattices, then f : L + L’ is called a lattice homomorphism (lattice dud homomorphism) if it is both a meet and a join homomorphism

(if f(. A Y) = f ( 4 v f ( Y ) and f ( x v Y) = fb)A f(Y)). 1.21. Definition.

Let

A

P

P’ be posets and f : P -+ I”. If S is a subset of P such that x exists ( V z exists) then f is said t o preserve the meet A x and

XES

XES

provided

A

XES

join

V x

XES

XES

f ( x ) exists and provided

f( A

V f(x)

XES

x) =

exists and f (

XES

L e t rn be an infinite cardinal. Then

A

f(x) (to preserve the

V

x) = V

XES

XES

f

XES

f(x)).

is called an m-complete homo-

morphism or simply an rn-homomorphism provided it preserves all existing meets and joins of sets of cardinality a t most m . We say that f is a complete

homomorphism if it is an m-homomorphism for every m, i.e. if is preserves all existing meets and joins. The map f is said to be an rn-complete dual homomorphism (a complete dual homomorphism) provided it is an rn-complete homomorphism (a complete homomorphism) from P t o the dual of PI. If P and P’ are lattices it may be convenient t o replace, in the above terms, t he word “homomorphism” by the words “lattice homomorphism” (this is consistent with Definition 1.21).

Lattices, universal algebra and categories

10 1.22. Remark.

Every semilattice (lattice, m-complete, complete) homomorphism is isotone and every semilattice (lattice, m-complete, complete) dual homomorphism is antitone, because

and dually. In the case of mappings between tosets, isotone (antitone) map-

pings coincide with lattice homomorphisms (lattice dual homomorphisms) by an argument similar t o t h e above one.

A t this point it seems a good idea to introduce the concepts of meet, j o i n , lattice, rn-complete lattice and complete lattice isomorphism, according t o a general usage in algebra. Thus e.g. a meet isomorphism is a bijective

meet homomorphism, which implies t h a t th e inverse mapping is also a meet homomorphism; etc. However, the novelty is illusory: 1.23. Proposition. (i)

Every isomorphism in the sense of Definition 1.18preserves all ezisting meets and joins.

(ii) T h e concepts of meet, j o i n , lattice, m-complete lattice and complete lattice isomorphism reduce t o the concept of isomorphism in Definition 1.18, while the dual concepts reduce to the concept of dual isomorphism in Definition 1.18.

Proof. (i) Suppose f : P + P’ is a poset isomorphism, S C P and a = exists. Then for every

5

E S, from a

Ix

we get f(a)

A x

XES

5 f(x). If

b‘ E P’ fulfils b’ 5 f(x) for every x E S then since b’ = f(b) for some b E P , we apply f-’ t o f ( b ) 5 f(x) and get b I 5 for every 2 E S,

Posets and lattices

11

(ii) Take e.g. a meet isomorphism f : L -+ L'. Then f and f-' are meet homomorphisms, hence they are isotone, therefore

f

isomorphism. The converse follows from (i).

is a poset 0

In view of Proposition 1.23 th e terms isomorphism and dual isomorp-

h i s m can be unambiguously used without further specification, as well as the notation L

L'.

1.24. Definition.

P into a poset P' is a mapping f : P + P' such that the corestriction f : P -+ f ( P ) is an isomorphism. An embedding of a poset

1.25. Remark.

The map f is an embedding iff

because (1.21) implies f(x) = f(y)

+ I = y.

Necessity is trivial.

According t o a general usage in mathematics we are interested in embedding an arbitrary poset into a poset having a much more particular structure.

A typical example is the next theorem, known as the MacNeille completion by cuts: 1.26. Theorem. For every poset P , the m a p (1.22)

z

H

I(.

=

(2

EP

12

5 x}

is an embedding of P into the complete lattice

M

(1.23)

={X

P I X = X+-}

(6. Example 1.17). This embedding preserves all meets and joins existing in P . Proof.

M

is a complete lattice by Remark 1.16. For every z E P we have

Lattices, universal algebra and categories

12 {z}+ =

{t E P 1 z 5 t } , hence (z] = { z } + - E M . Clearly z 5 y

H (z] C_

(y], therefore the map (1.22) is an embedding by Remark 1.25. For the second part of the theorem we compute meets and joins in

1.16 w i t h ’ M =

M

and

M

via Remark

L = P(P).Note that inf and sup in L are

set-

theoretical intersection and union, respectively, hence it remains t o prove that

(1.24”)

provided the left-hand sides exist. But (1.24’) is a mere translation of the

5 z (Vz E X ) . Further set u = V X and Y = U{(z] 12 E X } . Then X Y , hence Y + E Xt and since a E Xt it follows t h a t every y E Y+ fulfils a 5 y, therefore z 5 y for every y E Y + and every z 5 a ; in other words ( u ] C Y + - . On t h e other hand every y E Y fulfils y 5 z for some z E X , hence y 5 a ; it follows that a E Y + ,therefore 0 Yt-E { u } - = (u]. Thus ( u ] = Y + - completing the proof. obvious property z

5

AX H z

1.27. Definition. Every meet (join, lattice, dual lattice) homomorphism of the form f : L -+

L is called a meet (join, lattice, dual Eattice) endomorphism. Every (dual) isomorphism of t h e form f : L -+ L is called a (dual) automorphism. 1.28. Definitio n. An inuolutive operation or simply an involution of a set X is a mapping

f :

X

+ X such that f ( f ( z ) ) = 2 for all z E

X.

1.29. Remark.

A mapping f f-l

:

X

--f

X

is an involution if and only if f is a bijection and

= f. An increasing (a decreasing) involution on a toset is an automorp-

hism (a dual automorphism).

13

Posets and lattices From now on we shall often use notations like f z instead of f(z). 1.30. Proposition.

Let N be a decreasing involution on a meet semilattice ( L ,A) (join semilattice ( L ,V ) ) . If one defines V b y (1.25’)

z V y = N ( N z A Nu)

then L becomes a lattice ( L ,A, V ) in which

holds. (If one defines A b y (1.25”) then L becomes a lattice ( L , A , V ) in which (1.25’) holds). Proof. Suppose ( L ,A) is a meet semilattice and show that N ( N z A Ny) = 1.u.b. { z , y } . But Nz A Ny 5 Nz implies z = N N z 5 N ( N z A Ny) and similarly y 5 N ( N z A N y ) . Further if z 5 z and y 5 z then N z 5 N z and

N z 5 N y , hence N z 5 Nz A Ny, therefore N ( N x A Ny) 5 N N z = z . Thus ( L ,A, V ) is a lattice and (1.25”) follows from Remark 1.29.

0

Let further ( L ,A, V ) be a lattice. According to Definitions 1.27 and 1.28, an involutive dual endomorphism of L is a mapping N : L --t L such that

,

(1.26)

NNz =x

(1.27’)

N ( z A y) = Nz V Ny

,

(1.27”)

N ( x V y) = N X A Ny

.

1.31. Proposition. Let ( L , A , V ) be a lattice and N : L equivalent:

(i)

L. The following conditions are

N is a n involutive dual endomorphism;

(ii’) N fuIfils (1.26) and (1.27’); (ii”) N fulfils (1.26) and (1.27”);

Lattices, universal algebra and categories

14

(iii’) N fulf;Zs (1.25’); (iii”) N fulf;ls (1.25”). Proof.

(i) j (iii’): Immediate. (iii’) j (ii’): N N z = N ( N s A N c ) = x V z = x, then NxVNy = N(”z A NNy) = N ( e A y). (ii’) j (i): N z A N y = N N ( N c h N y ) = N(NNsVNNy)= N ( z V y ) . 0

The proof is completed by duality.

1.32. Proposition. L e t P be a p o s e t and d :

P

+ P a decreasing m a p p i n g .

(i) If P h a s least e l e m e n t 0 (greatest e l e m e n t 1) a n d d i s surjective, t h e n dO is greatest e l e m e n t ( d l i s least e l e m e n t ) . (ii)

If P =

{al, ..., anel} with al

da; = a,-i

(i = 1 , ...,n - 1).

< ... <

a n d d is injective, t h e n

Proof.

(i) Every x E

P

can be written z = dy for some y E P , hence 0

implies z = dy

5 do.

(ii) It follows from the hypotheses that the single possibility is da,-l = al, an-1.

5

y

< ... < daz < d a l , hence = az, ..., da2 = dul = 0

The next point of this section is the general concept o f subalgebra applied t o lattice theory.

1.33, Definition.

A subsemilattice of a meet semilattice ( L ,A) (join semilattice ( L ,V)) is a subset S C L such that c,y E S + z A y E S (c,y E S + s V y E S).A sublattice of a lattice ( L ,A, V) is a subset S L that is both a subsemilattice of ( L ,A) and of (L, V).

15

Posets and lattices

1.34. Remark. A subsemilattice (sublattice) S of L is itself a semilattice (lattice) with respect t o the restriction(s) t o S of the operation(s) of L . However, a subset S 5 L which is a semilattice (lattice) with respect t o t h e partial order inhe-

L

rited from

L ; see e.g. be a subsemilattice of (L,V).

need not be a subsemilattice (sublattice) of

1.16 in which M need not

Remark

We conclude this section with an alternative description of the MacNeille construction in Theorem 1.26. This variant, obtained with the aid of an isomorphism, justifies t h e term “completion by cuts” by analogy with the Dedeking construction of the irrationals as cuts of the set of rational numbers. Consider again the completion

M

such that

M

P C L. The bijection y

of P and construct a bijective copy :

M

L of

+ L (where ~ ( ( x ]= ) x) makes

L into a lattice isomorphic t o M via th e definition y ( X ) 5 y ( Y ) e X C_ Y. For every x E L set A, = ( a E P 1 a 5 x}. 1.35. Lemma.

FOT every X E M and x E L:

(i)

y ( X ) = supLX ;

(ii) cp-’(s) = A, ; (iii) z = supLA, ; (iv)

A,+ = { b E P 1 z

Ib}.

Proof. The obvious equivalence ( u E

X+-E M

A- e ( a ] C A - ) applied t o X =

yields

(1.28)

X = U

(1.29)

X

aEX

= {u E

(u]

,

P I ( a ] C X} .

p(X) = x. Then we obtain (i) by applying y to (1.28). Further, since ( u ] C X @ u 5 x we can write (1.29) in the form X = A,, which i s Set

Lattices, universal algebra and categories

16

(ii). From (i) and (ii) we obtain (iii). Finally we prove (iv): if x 5 b E P then b E A,+ by the definition of A,; conversely, if b E A,+ then b E P and 0 5 5 b by (iii). 1.36. Lemma. If P = T is a chain then A, U Af = T for every x E L. Proof. If a E T

- A,

then a $ x, hence x

< a , therefore a € Af

1.35(iv).

by Lemma 0

1.37. Definition. A cut of a chain T is a pair of sets ( A , B ) such that

(i) a

< b (Va E A ) (Vb E B ) ,

(ii) A U B = T , (iii) A has no sup in T and (iv)

B has no inf in T.

1.38. Proposition. Let L be the MacNeille completion of a chain T . Then: (a) L is a chain;

(b) the map x H (A,, A f ) establishes a bijection between L - T and the cuts of T , in which the inverse image of a cut (A, B ) is x = supLA = infLB, characterized b y a < 2 < b (Vu E A ) (Vb E B ) . Proof. (a) Let x,y E

L. If A, C A,

then

x 5

y. Otherwise there exists a E

A,-A,, hence a E A, and a E A: by Lemma 1.36, therefore y 5 a 5 z. (b) Let (i)-(iv) stand for the properties in Lemma 1.35.

L - 2'. For every z 5 b by (iv); but

First we prove that (A,,Af) is a cut for every x E

a E A, and b E A: we have a

5x

by (iii) and

17

Posets and lattices

a , b E T therefore a < x < b. Further A, U A: = T by Lemma 1.36. The existence of a = supT A, would imply t h e contradiction x = a E T by (iii) and Theorem 1.26. The existence of b = infTA: would imply a 5 b (Va E A,) (because a 5 b‘ (Vb‘ E A:)) i.e. b E A:, hence b would be the least element of A: i.e. b = supT A,, which is impossible. The map x

H

(A,,Af)

is injective by (ii).

If (A,B) is a cut set 2 = supLA. Then 2 E L - T . If a E A, then a E A, because otherwise a E B hence a’ 5 a (Va’ E A ) , therefore 2 5 a , consequently x = a E T , a contradiction. Thus A, E A and since the converse also holds by (iii) it follows that A = A,. If b E B then b E T and a < b (Va E A ) , hence 5 5 b by (iii),therefore b E A,+ by (iv); we have thus proved that B C A .: To prove the converse take b E A: i.e. b E T and x 5 b by (iv); then b E B because otherwise b E A hence b 5 2 therefore x = b E T , a contradiction. Therefore Af C B , consequently B = A:. We have thus proved that (A,B) = (A,,A,+) for 5 E L - T given by x = supL A, = infL A:; clearly a < x < b (Vu E A) (Vb E B ) .

L satisfies a < y < b then by x = supL A and y 5 x by x = infL B therefore y = x. 0

Finally if ( A , B )= (A,,AZ)

x 5y

is a cut and y E

Lattices, universal algebra and categories

18

$2. Distributive lattices, De Morgan and Boolean algebras T h e s t a c t u r e s indicated i n t h e title are successive specializations of lattices. We introduce them here in view of their importance and as prerequisites t o t h e study of Lukasiewicz-Moisil algebras, which are situated between De Morgan algebras and Boolean algebras.

2.1. Proposition.

The following identities and implications are equivalent in a lattice: (2.1’)

x A ( y V z ) = (x A y ) V ( a : A z ) ;

(2.1”)

x V (y A Z) = (X V y) A (X V Z) ;

(2.2’)

z A (y V z )

(2.2”)

x V (y A z ) 2 ( x V y) A ( x V z ) ;

(2.3)

(x A y ) V (y A Z) V (x A z ) = (z V y ) A (y V z ) A ( x V z ) ;

(2.4)

x Az 5 y &x

(2.5)

xAz 5 y A z &xVz 5 y V z j x

(2.6)

x A z =~

5 ( x A y) V ( x A z ) ;

I y V z J x 5 y;

A &z x V z = y V z

j

5 y;

x =y.

Proof.

(2.1’) H (2.2’): For (x A y) V (X A z ) 5 x A (y V z ) in any lattice. (2.1’)

J

(2.3): Calculate successively

(x

v 9) A (Y v 2) =

((x

v y) A Y) v

((x

v Y) A 2) =

= yV(zAz)V(yAz)=yV(xAz) (x V y) A (y V

Z)

A

(X

,

V Z ) = ((x V z ) A y) V ((x V z ) A z A z ) =

= ( x A y) V ( z A y) V (x A z ) .

(2.3)

j

and x V z

(2.4): Suppose z A z 5 y and z 5 y V z. Then z A z I yA z

5 y V z , hence

19

D i s t r i b u t i v e lattices, D e Morgan and Boolean algebras

=

(2

A y) V (x A z ) V (y A Z) =

-

= ( ~ A Y ) V ( Y A ~ ) I Y

(2.4)

+ (2.5):

If x A z 5 y A z and x V z 5 y V z then x A z

5 y and

x
(2.5) =+ (2.6): Obvious. (2.6) +- (2.1’): We first prove that the lattice is modular, i.e. (2.7’)

x

(y V Z) = (x A y) V z

,

(2.7”)

x 5 z + x V (y A 2) = (x V y) A z

.

2 z jx A

Take x, y, z such that x 2 z and set a = x A (y V z ) , b = (x A y) V z , c = y.

Then b 5 a, hence bAc
1.e.

aAc=bAc.

Also

aVc> bvc

1.e.

aVc= bVc.

2 z V y =y V z V y 2 a V c ,

Therefore a = b; i.e. (2.7’) holds and by duality we get (2.7”). Now take arbitrary 2, y, z and set a = ( X V y) A (Z V

b = (9 V z ) A

(x A y)) = ((x V y) A z ) V (x A y) ,

(X V (y A 2))

= ((9 V z ) A x) V (y A Z)

,

where we have used (2.7), and c = y. Then ~ A c y= A ( z V ( x A y ) ) = ( y A z ) V ( z A y ) ,

bAc= yA(zV(yAz))

=(yAx)V(yAz),

hence a A c = b A c and similarly a V c = b V c . It follows that a = b, therefore x A b = x A a i.e. x A (y v z > = x A ( z v (x A y)) = (x A z ) v (x A y). Thus (2.1’), (2.2’) and (2.3)-(2.6) are equivalent. The proof is completed by duality. 0

Lattices, universal algebra and categories

20 2.2. Definition.

A lattice satisfying the equivalent conditions i n Proposition 2.1 is said

to be

distributive. 2.3. Example. Every toset is a distributive lattice. Here

tA

y = min(z, y) and t V y =

rnax(z, y), i.e. the least and greatest element of t h e set

(5,y},

respectively.

As a matter of fact all th e lattices studied in this book are distributive. However, non-distributive lattices do exist, e.g.:

1) t he pentagone, i.e. th e five-element lattice (0, a , b, c, l} where a # b and b # c while a < c; 2) t he diamond, i.e. the five-element lattice having three pairwise incomparable elements;

3) t he lattice of normal subgroups of a group, where t h e partial order is set-t heoretical inclusion. Note t hat t he pentagone is not even modular, i n contrast with t h e other t w o lattices. T h e next tw o short characterizations of distributive lattices t u r n o u t t o be very useful.

2.4. Proposition (Sholander [1951]).

Let L be a set and A, V : L2 L. Then ( L ,A , V) is a distributive lattice i f and only if the following identities are satisfied: --f

V 8) =t

,

(2.8)

2

A

(2.9)

2

A (y V z ) = ( z A

(Z

t)V

(y A z) .

Proof. Necessity is trivial. To establish sufficiency we first prove idempotency in the following steps:

21

Distributive lattices, De Morgan and Boolean algebras z A z = (z A z) A

((2

A z) V (z A z)) = (Z A z) A

5

,

zAz = z A ((zj\z)V(~Az)) =

= ((z A X ) A Z) V ((z A

z v 2 = ( X A X ) v (ZA 3) = 5

Z)

A z) = (Z A X ) V (z A Z) = z

,

.

Further we prove commutativity and the dual absorption law in four steps:

z A Y = Z A ( Y V Y ) = ( Y A Z ) V ( ~ A Z )= y~

5 ,

(zAy)Vz =(yAz)V(a:Az) =z A ( z V y ) =z xA(yVz)=

=

,

(zAz)V(yAz)=~V(zAy) = 2

V ((z A y) A ((z A y) V

3))

=

= (z A X ) V ((z A y ) A x ) = = ( z A ((zAy)Vz) = z A z = z , z V y = (z A (y V z)) V (y A (y V z))

=

= (yVz)A(yVz)=yVz. The steps for associativity are as follows: zA((zVy)Vz) z

v (Y v 4

(ZV y) V

=

= (zA(zVy))V(sAz)=zV(zAz)=z,

(.

A

v

(2 A

((. v Y) v 2)) v ((Y A ((Y v 4 v 2)) v

v Y) v 2)))

=

v Y) v 2)) v

(((z

((z

=

(. A ((.

=

(bv Y) v z ) (. v (9 v 4)

v Y) v z )

(Y v 4 ) =

9

z = z V (3 V Z) = ((z V y) V z) A =

A

((3

(Z V (y V z))

=

V y) V z ) A (z V (y V z)) = z V (y V z )

and the other associative law follows dually.

0

Lattices, universal algebra and categories

22

2.5. Proposition (Ferentinou-Nicolacopoulou [1968]). Let L be a set, 0 E L and A,V : L2 + L. Then (L,A,v,O)i s a d i s t r i ~ u t i ~lattice e with zero if and only if it satisfies identities (2.8) and

(2.10)

x A (y V z ) = ( z A (x V 0)) V (y A (x V 0))

.

Proof. Follows from Proposition 2.4 provided we succeed to show that xV0 = x. But x V z = (xA(xVO))V(XA(XVO))

=zA(zV~)=z,

XAX=XA(XVX)=X,

xAy=xA(yVy)

= (yA(zV0)) V (yA(zV0))

=

= yA(zVO), x

v o=(zvO)A(X

vO) = X

A ( X

vO) = 2 .

2.6. Defi n ition. An algebra ( L ,A, V, N , 0 , l ) where ( L ,A, V, 0 , l ) is a bounded distributive lattice and N an involutive dual endomorphism of L, is said to be a D e Morgan algebra. N is called the negation or the involution of L . See Moisil (19351 and Kalman [1958]. See 1.27-1.31 for the previously defined concepts and alternative characterizations of N . See also Propositions 2.9-2.10. 2.7. Example. , U, 0,X ) Let X be a set and f : X + X an involution. The lattice ( P ( X ) n, endowed with the operation N X = X - f ( X ) , is a De Morgan algebra.

2.8. Example Every toset endowed with a decreasing involution is a De Morgan algebra; cf. Example 2.3 and Remark 1.22. As a matter of fact many lattices studied in this book are particular cases

of De Morgan algebras.

Distributive lattices, De Morgan and Boolean algebras 2.9. Proposition (Maronna [1964]). Let L be a set, A : L2-+ L and N : L (2.11)

tVy

=N(Nt A Ny)

-+

23

L. Define

.

T h e n ( L , A , V ) is a distributive lattice and N an involutive dual endomorphism of L if and only if the following identities are satisfied: (2.12)

z A N ( N t A Ny) = z

(2.13)

zA

N(Ny A N z ) = N ( N ( z A

Z)A N(y A z)) .

Proof. In view of (2.11), Axioms (2.12)-(2.13) reduce to (2.8)-(2.9), therefore Proposition 2.4 implies that (2.12)-(2.13) are necessary and sufficient for ( L , A , V ) to be a distributive lattice. Now N is an involutive dual endo0 morphism by Proposition 1.31, condition (iii'). 2.10. Proposition. Let L be a set, A : L2 + L, N : L --t L and 0 E L. Define V b y (2.11) and 1 = NO. T h e n ( L , A , V , N,O, 1) is a De Morgan algebra if and only if L satisfies identities (2.12) and (2.14)

tAN(NyANz) = = N(N(. A

N ( N A~ N O ) )A ~ ( A 3N ( N A~ N O ) ) ).

Proof. Similar to the proof of Proposition 2.9 but using Proposition 2.5 cl instead of 2.4, plus Proposition 1.32(i). 2.11. Definition. An element t of a bounded lattice is said t o be complemented or chrysippian if there is an element y such that t A y = 0 and t V y = 1; every element y with these properties is called a complement of 5. Note that if y is a complement of 5 then t is a complement of y. An element may have several complements, a single complement or none. Thus e.g. the element b of the pentagone has the complements a and c , while

Lattices, universal algebra and categories

24

each of the elements a and c has b as unique complement. Also, each of the three incomparable elements of the diamond has the two other elements as complements. It is important to note that in a bounded toset every element different from 0 and 1 has no complement.

2.12. Proposition.

(9

In every bounded lattice 1 is (0 i s ) the unique complement of 0 (of 1).

(ii) Every element of a bounded ddstrzbutive lattice has a t m o s t o n e complement. Proof.

(i) Obvious. (ii) Let y and y' be complements of z. Then

2.13. Definition. Let L be a bounded distributive lattice. We denote by C ( L ) the set of all complemented elements of L and call it the center of L . For each z E C ( L ) we denote by ii i t s unique complement. Thus

(2.15') z A 5 = 0 ,

(2.15") z V 5 = 1 . 2.14.Proposition. Let L be a bounded distributive lattice. T h e n 0,l E C ( L ) and f o r every x,y E C ( L ) it follows that x A y, x V y, 3 E C(L);namely,

25

Distributive lattices, De Morgan and Boolean algebras (2.16')

Sny = 5 V g ,

(2.16') (2.17)

=5A

-

x

X=

g,

.

Proof. 0 , l E C ( L ) by Proposition 2.12(i) and it remains to prove (2.16)(2.17). Using (2.1') and (2.1") we get ( X A Y ) A ( Z V ~ ) = ( X A Y A A ) V ( ~ A Y A ~=)

= (oAY)v(xAO)=OVO=O, ( ~ A y y ) v ( % V @=) ( z V 2 V y ) A ( y V S V j j )

=

= ( 1 ~ ~ ) ~ ( 1 ~ ~ ) = 1 ~ 1 = 1 ,

which implies (2.16'), while (2.16') follows by duality. Finally (2.15) yield D 5 A 5 = 0 and 5 V z = 1, that is (2.17). Proposition 2.14 minus (2.16)-(2.17) can be restated t o the effect that

C ( L ) is a sublattice of ( L ,A, V, 0 , l ) ; cf. Definition 1.33. 2.15. Remark. In a bounded distributive lattice

L , the following properties hold

for every

z E C ( L ) and y E L:

(2.18')

z A (5 V y) = z A y

,

(2.18')

.1:V

(5 A y) = 2 V y

,

(2.19')

x

2y

2 A y =0

(2.19')

t

5y

* 5 v y = 1,

,

because e.g. x A (3 V y) = (X A 2) V (z A y) = z A y and z 2 y implies 0 = Z A X 2 f A y , while i A y = 0 implies y = ( s V 5 ) A y = ( z A y ) V ( z A y ) = XAY. 2.16. Definition. A Boolean algebra is a bounded distributive lattice L such that L = C ( L ) ,

Lattices, universal algebra and categories

26

i.e. all the elements are complemented.

2.17. Examples. a) The two-element toset (0,1} is a Boolean algebra; cf. Example 2.3 and

Proposition 2.12(i), b) For every set X , the lattice ( P ( X ) ,n, U, 0 , X ) is a Boolean algebra. c) For every bounded distributive lattice

L , the sublattice C(L)is a Boolean

algebra by Proposition 2.14. This example is essential i n our book. 2.18. Remark. In view of Proposition 2.12(ii) and Definition 2.16, to every element x of a Boolean algebra L corresponds a unique complement 3 , so that taking complements is a unary operation - : L --t L. Therefore Definition 2.16 can be restated to the effect that a Boolean algebra is a De Morgan algebra

( L ,A, V,; 0 , l ) satisfying identities (2.15). Beside the properties established so far in this and the previous section, Boolean algebras have of course specific properties. Thus e.g. (2.20’)

x =y

(2.20”)

5

=y

($

( 5 A y) V

(x A

v) = 0 ,

( 3 V y) A (x V jj) = 1 ,

follow from (2.19). Also

(2.21’)

((u A x)V(bA

5)) A ( ( c A x) V ( d A 3 ) ) =

= ( U A C A ~ ) V ( ~ A ~ A Z ) ,

(2.21”)

( ( u A x)V(b A 5 ) ) V ((c A

x) V ( d A 3)) =

= ( ( u V c) A x) V ( ( b V d ) A Z) ,

(2.22)

( u A ~ ) v ( ~ A ~ ) = ( s ~ A ~ ) v ( ~ A ~ ) ,

are easily checked and can be generalized to several variables.

The following properties hold only in the Boolean algebra {O,l}:

27

Distributive lattices, De Morgan and Boolean algebras (2.23’)

xA y =0 e

x

(2.23”)

x v y = 1 e x = 1 or y = 1 .

= 0 or y = 0 ,

Recall that;according t o Definition 1.21, if L and L‘ are lattices, a mapping f : L + L’ is called a lattice homomorphism provided it preserves A

and V, i.e. f(x A y) = f(s)A f(y) and f(z V y) = f(x) V f(y); the Same concept applies in particular t o distributive lattices.

2.19. Definition. If L and L’ are bounded lattices, a lattice homomorphism f : L

-+

L’ is

said t o be a bounded-lattice homomorphism provided it preserves 0 and 1, i.e. f(0) = 0 and f(1) = 1. If, moreover, L and L‘ are De Morgan algebras and f preserves

N , i.e. f ( N s ) = N f ( s ) , then f is termed a De Morgan

homomorphism.

2.20. Remark. Let L and L‘ be De Morgan algebras. Every meet (join) homomorphism

f :L

+ L’ that preserves N , 0 and 1 is a De Morgan homomorphism. For

2.21. Remark.

L‘ be bounded distributive lattices. Every bounded-lattice homomorphism f : L -+ L‘ preserves complements whenever they exist, i.e.

Let L and f(5) =

m.

For it is easily seen that f ( x ) A f ( Z ) = 0 and f(x)Vf(.)

= 1.

2.22. Definition.

Let L and L’ be Boolean algebras. A mapping f : Boolean homomorphism provided it preserves A, V,;

L

L’ is called a 0 and 1. In other +

words, a Boolean homomorphism is a De Morgan homomorphism between Boolean algebras.

28

Lattices, universal algebra and categories

2.23. Proposition. Let L and L‘ be Boolean algebras and f conditions are equiwalent:

:

(i)

f is a Boolean homomorphism;

(ii)

f is a bounded-lattice homomorphism;

L + L’. T h e following

(iii’) f is a meet h o m o m o r p h i s m that preserves complements; (iii”) f is a j o i n h o m o m o r p h i s m that preserves complements. Proof.

(ii) j (i): f(z)Af(5) = f(zA5) = f(0) = 0 and similarly f(z)Vf(5) = 1, hence f(5) =

fo.

(iii’) + (i): f is a j o i n homomorphism as in Remark 2.20. Then f(0) = f ( z A 5 ) = f ( z ) A f o = 0 and similarly f(1) = 1. T h e converse implications are trivial.

0

W e conclude this section w i t h a few words about subalgebras.

2.24. Definition. A 0-1-sublattice of a bounded lattice (L, A, V, 0 , l ) is a sublattice S of L such t h a t 0 E S and 1E S. A D e Morgan subalgebra of a D e Morgan algebra ( L ,A, V, N , 0 , l ) is a 0-1-sublattice S of L such t h a t z E S

+ N z E S.

A Boolean subalgebra of a Boolean algebra ( B ,A, V,; 0 , l ) is a D e Morgan subalgebra of B viewed as a Boolean algebra. Clearly a 0-1-sublattice of a bounded (distributive) lattice is itself a bounded (distributive) lattice, while a De Morgan (Boolean) subalgebra of a De Morgan (Boolean) algebra is itself a De Morgan (Boolean) algebra. T h e converses do not hold, as can be seen e.g. f r o m the following example, which is important in many respects.

29

Distributive lattices, De Morgan and Boolean algebras 2.25. Example.

L be a

Let

(distributive) lattice and take a, b E L with a

5 b.

The segment

or closed interval [a,b] defined by (2.24)

[a, b] =

is a sublattice of

{z E L I a _< z _< b}

L

,

and a bounded (distributive) lattice with respect t o the

L. However, if L is bounded but a # 0 or b # 1 then [a,b] is not a 0-1-sublattice of L. In particular if L is a Boolean algebra then

order inherited from

[a,b]is not a Boolean subalgebra, yet if we define (2.25)

d = (2 V a ) A b = ( 3 A b) V a

(vz E [a,b ] ) ,

then ( [ a ,b],A, V,', a, b) is a Boolean algebra; the proof is easy.

2.26. Proposition. Let L be a De Morgan algebra and ( u , ) , ~ sC L. Then (2.26')

N

(2.26")

N

v

A a, =

sES

SES

Nus,

V a, = A N u , , SES

SES

to the effect that the existence of one side of the equality implies the existence of the other side and the equality itsey. Proof. We suppose a =

A a,

sES

V Nu,.

8ES

But a

(Vs E S). Then a,

I a,

hence N u

2 Nb

Thus the existence of

V a,

V Nu,,

(Vs E

S). Further t a k e b 2 N a ,

2 Nb, therefore b 2 N u .

A a, implies (2.26')

and by duality the existence

implies (2.26"). These results can be applied t o

A Nu,

and

,€S

s€S

sES

2 Nu,

(Vs E S), hence a SES

of

and check t h a t N u fulfils the definition of

completing the proof.

0

Lattices, universd algebra and categories

30

Proposition 2.27 below, as well as Remark 2.13, generalizes well-known properties of Boolean algebras. 2.27. Proposition. Let L be a bounded distributive lattice, (a,),€, G L and c E C ( L ) . Then (2.27')

v

a, exists

=$

A

a, exists

+

SES

(2.27")

SES

cA

cA

V a,

V

( c A a,)

,

V a, = A

(cV a,)

,

SES

SES

=

SES

SES

to the effect that the right sides exist and the equalities hold. Proof. Suppose a =

V

SES

a , exists. Then a 2 a , hence c A a

2 c A a,

S ) . Now take b 2 c A a , (Vs E S ) ; then E V b 2 C V a, (Vs E S) by (2.18'), hence C V b 2 C V a , therefore b 2 c A b 2 c A a by (2.18'). Thus (Vs E

cAa=

V

(cAa,).

0

SES

2.28. CorolIa ry .

I n a n y Boolean algebra Properties (2.27) hold for arbitrary elements c, a, (s E S).

Filters, ideals and congruences in lattices

31

$3. Filters, ideals and congruences in lattices

The dual concepts of filter and ideal t u r n o u t to be a powerful t o o l o f lattice theory, w i t h applications i n other fields of mathematics. T h e main points of this section are t h e complete lattices of all filters and of all ideals, the characterization of prime filters and o f prime ideals, the relationship between prime and maximal filters (ideals), t h e congruences associated w i t h filters. W e refer t o distributive lattices and t o Boolean algebras; t h e results will be generalized in Chapter 5 @l-Z to the framework of Lukasiewicz-Moisil algebras.

3.1. Definition. Let

L

b e a lattice. A non-empty subset

F C L ( I 2 L ) is called a fiEter (an

i d e a l ) provided for every z,y E L ,

+x A y E F , & x 5 y +y E F

(3.1)

z,y E F

(3.2)

z EF

(for every z,y E

L,

(3.1')

z , y ~ l + ; c V y ~ I ,

(3.2')

x ~ l & y l z + y ~ I ) .

3.2. Remark. The concepts o f filter and ideal are dual t o each other. Therefore w i t h each theorem and definition referring to filters is associated a dual theorem and a dual definition, respectively, referring to ideals. In view of t h e duality principle we can restrict to one proof for each pair of dual propositions.

3.3. Examdes. a) The lattice L itself is b o t h a filter and an ideal, called t h e i m p r o p e r filter

( i d e a l ) , while the other filters (ideals) are said to be proper. b) For every u E L, the set [u ) = {z E L I u 5 z } is a filter (the set (a]= { z E L I z 5 u } is an ideal) called t h e p r i n c i p a l f i l t e r ( i d e a l )

Lattices, universal algebra and categories

32

L we denote by generated b y a. More generally, for each subset X [ X )(by ( X I ) the filter ( i d e a l ) generated b y X , i.e. the least filter (ideal) which includes X ; the term "least" refers t o set inclusion G as partial order. Example 3.3.b requires a proof which we provide below. In the sequel we shall generally leave t o the reader the statements of the dual theorems and, of course, their proofs; cf. Remark 3.2. Caution: the duality principle refers to the lattice-theoretical

concepts and not t o t h e set-theoretical

ones; see

e.g. the dual definitions in Example 3.3.b. 3.4. Proposition.

Let L be a lattice and X G L.

If X # 0 t h e n

and if L has greatest e2ement 1 t h e n

[0) = 1.

Proof. Let Y stand for the right side of (3.3). Clearly X Properties (3.1) and (3.2) are easily verified, therefore

E Y ,hence Y # 0. Y is a filter. Sup-

F . If y E Y then z1A ... A z, < y for some zl,...,5, E F ,therefore z1A ... A t, E F by (3.1) and this implies y E F by (3.2). Thus Y E F completing the proof of Y = [ X ) . The reader is pose

F is a filter and X

urged t o check that (1) is the least filter, which i s equivalent t o (1) = [0).0 Numerous filters and ideals are dealt with in this book, where they play an important role. 3.5. Remarks. a)

A non-empty subset F G L is a filter if and only if (3.4)

z E F & y E F + z A y E F .

b) Let F be a filter of a lattice L. If L has greatest element 1then 1E F . If L. has least element 0 then 0 E F if and only if F = L.

Filters, ideals and congruences in iat tices

33

3.6. Theorem. If the lattice L has least element 0 (greatest element 1) t h e n t h e ideals ((Fl(L), G)) and the m a p (filters) of L f o r m a complete lattice (Id(L), x H (x] (x H [x)) is a n injective lattice (dual lattice) homomorphism.

c)

Proof of the filter part. FL(L) is a Moore family on

( P ( L )5 , ) by

Proposi-

tion 3.4; cf. Definition 1.14. Now apply Remark 1.16. Finally it is easy t o check that

[xV y) = [x)n [y) and [x A y) = [x)V [y).

0

3.7. Definition.

A proper filter F (proper ideal I) is said to be p r i m e if for every 2,y E L , (3.5)

X V ~ E F J Z EorFy e F

(for every z , y E L , (3.5')

xAyEI+zEI

or y E I )

For example every proper filter (ideal) in a chain is prime. Further examples will be the object of various propositions of this book.

3.8. Remark. If P is a filter (proper filter, prime filter) of a De Morgan algebra L , then N P = {NxIx E P} is an ideal (a proper ideal, a prime ideal) of L. 3.9. Proposition. T h e following conditions are equivalent f o r a subset F of a lattice L:

(i) F is a prime filter; (ii) F is a proper filter and f o r every x,y E L ,

(iii) L - F is a prime ideat (iv) there i s a surjective lattice h o m o m o r p h i s m h : L -+

that F = h-'({I}).

{0,1} such

Lattices, universal algebra and categories

34 Proof.

(i)+ (ii): By Definition 3.7 and (3.2).

(ii) + (iv): Set h ( z ) = 1 for z E F and h ( z ) = 0 for z E L - F. Then h is surjective because 0 # F # L and it follows from (3.4) that

h ( z A y ) = 1 @ h ( z ) = 1 & h ( y ) = 1 M h ( z )A h ( y ) = 1 , therefore h ( z A y ) = h ( z ) A h ( y ) .Similarly one proves h ( z V y ) = h ( z ) V h ( y ) , using (3.6). (iv) =+ (ii):

0# F # L

because

h is surjective. Then

X Ay E F M h ( x ) A h ( y ) = h ( z A y ) = 1

5

E

F &yE F ,

therefore F is a proper filter by Remark 3.5.a and one proves similarly (3.6).

(ii) j (i): Trivial. Thus (i) @ (ii) M (iv) and it follows by duality t h a t (iii) is equivalent t o t he existence of a surjective homomorphism h :

L

+

{0,1} such that

L - F = h-’({O}); but the latter property is equivalent t o (iv).

0

3.10. Theorem. Let L be a distributive lattice, F afilter and I an ideal of L. If F n I = 8 then there is a prime filter P such that F P and P f l I = 0.

c

F’ which satisfy F c F’ and F’ n I = 0. It follows from t h e Zorn lemma that G has a maximal element P . Since P E G it remains to prove t h a t the filter P is prime. P is proper because P n I = 0. Suppose P is not prime. Then there exist a , b E L such that a V b E P , a # P and b # P . Let X = P U { a } . Then [ X )n I # 0, otherwise P c [ X ) E G contradicting the maximality of P . Take z E [ X ) n I .Then Proposition 3.4 implies easily the existence of p E P such that p A a 5 z and since z E I it follows that p A a E I . Similarly there is q E P such that q A b E I . Then ( p A a ) V ( q A b ) E I and on t h e other Proof. Let G be the family of those filters

hand

Filters, ideals and congruences in lattices therefore I n

P # 0, a

contradiction.

In Corollaries 3.11-3.14

3.11. Corollarv. If I is an ideal and a E L and P n I = 0. Proof. Take

F = [a)

35 0

L is a distributive lattice.

-I

there is a prime filter P such that a E P

in Theorem 3.10.

0

3.12. Corollary. If F is a filter and b E L - F there is a prime filter P such that F C P and b # P . Proof. Take

I = (b] in Theorem 3.10.

0

3.13. Corollarv.

If a, b E L are such that a and b # P .

b there is a prime filter P such that a E P

Proof. Take I = ( b ] in Corollary 3.11.

0

3.14. Corollarv.

In a distributive lattice every proper filter is included in a prime filter and is an intersection of prime filters.

F be a proper filter. Denote by 3 the non-empty (cf. Corollary 3.12) family of those prime filters that include F . We set F' = n { P I P E 3} and prove that F = F'. Otherwise there is a E F ' - F , hence Corollary 3.12 yields a prime filter P E F s u c h that a # P,which contradicts a E F'.O Proof. Let

3.15. Definition. By a mazimal filter (mazimal ideal) is meant a maximal element of the family of all proper filters (proper ideals) ordered by set-inclusion. A maximal filter is also called a ultrafilter.

Lattices, universal algebra and categories

36 3.16. Proposition.

In a distributive lattice every maximal filter is prime. Proof. Let F be a maximal filter. Then F is included in a prime filter Corollary 3.12, therefore

P

F = P by the maximality of F .

by 0

Now let us turn t o Boolean algebras. First we extend Remark 3.8 and the notation N P . For every subset A

of a Boolean algebra B set A = {5 I z E A } .

3.17. Lemma. T h e following conditions are equivalent f o r a filter F of a Boolean algebra

B: (i)

F is proper;

(ii) F n F = 8 ; (iii) f o r every z E B at m o s t one of the elements

5, 5

belongs t o F .

Proof.

(i) + (iii): Otherwise 0 = z A 5 E F hence F = B. (iii) + (ii): Otherwise there is z E F n F , hence 2 E F and some y E F , therefore a: = y E F . (ii) + (i): Trivial.

2

=

for 0

3.18. Proposition.

T h e following conditions are equivalent f o r a filter F of a Boolean algebra

B: (i) F is mazimal;

(ii) F is prime;

(iii) f o r every z E B , exactly one of the elements Proof.

(i) + (ii): By Proposition 3.16.

2,

5 belongs t o F .

Filters, ideals and congruences in lattices

37

(ii) + (iii): x V ii! = 1 E F hence x E F or ii! E F , but not both because of Lemma 3.17. (iii) + (i): Note first that F is proper. If G is a filter such that F C G then there exists x E G - F , hence x E G and 2 E F c G, therefore 0 0 = x A f E G showing that G = B . Proposition 3.18 has t h e following converse:

3.19. Theorem (Nachbin [1947]). A bounded distributive lattice L is a Boolean algebra if and only if every prime filter of L is maximal.

if” part is included in Proposition 3.18. For the “if” part suppose t he lattice L contains an uncomplemented element a . Consider the filters FO = {x E L I a V x = 1) and Fl = {x E L I a A y 5 x for some y E Fo}. Then 0 4 F1 otherwise a would be complemented; hence there is a prime filter Pl such t h a t Fl C Pl.Let I = ( ( L- Pl)U { a } ] . Note that L - PI C I because a E I and a E Fl Pl implies a # L - Pl.Further we prove that Fo n I = 8. Otherwise t a k e x E Fo n I. Then x E Fo and since L - PI i s an ideal by Proposition 3.9, the dual of Proposition 3.4 implies easily that x 5 a V y for some y E L - Pl.Then 1 = a V x 5 a V y therefore y E Fo G Fl E P,a contradiction. Thus Fo n I = 0 and Theorem 3.10 yields a prime filter P such that Fo E P and P n I = 0. It follows that 0 P C L - I c Pl, therefore P is not maximal. Proof. The “only

The concept of congruence in a lattice is a specialization of t h e general notion in Definition 5.11: a congruence on ( L ,A, V) is an equivalence relation p on

L such that x p y & x’py’ =+x A x’py A y’ & x V x‘py V y’

(3.7)

For example, the relations ker

h :L (3.8)

--t

h, L2 and AL

L’ is a homomorphism, z ker h y H h(x) = h(y)

,

.

are congruences, where

Lattices, universal algebra and categories

38

(3.9)

x L2y vc, y E

L

,

(3.10)

XALY x = y

.

*

3.20. Definition. Let DO1 be the class of bounded distributive lattices. For every L E DO1 and every filter F of L , let m o d F be the relation on L defined by (3.11)

xmodFy@3uEFxAu=yAu

3.21. Proposition.

Let L E D01. T h e n m o d F is a congruence o n L for every filter F of L, and F = { x E L I z m o d F l } . Proof. Easy via the following remark.

If x A u = y A u and x' A u' = y' A u'

withu,u'€ F t h e n x A v = y A v a n d x ' A v = y ' A v w h e r e v = u A u ' € F . Forthesecond p a r t x E F ~ x A x = i A x + x m o d F l a n d x m o d F l ~

3u E F x A u = 1 A u

+ x 2 U , u E F + x E F.

0

3.22. Corollary. Suppose L E DO1 and F is a proper filter of L. Let L/F stand f o r LfmodF. T h e n L / F E DO1 with least and greatest elements 6 and i = F , respectively, and card(L/F) > 1.

3.23. Proposition. T h e following conditions are equivalent f o r L E D01, card L > 1, and a filter F of L:

(i) t h e only filters of L / F are the trivial filters

{i}and L / F ;

(ii) F is mazimal. Proof. l(ii)

+ l(i): Suppose F is not maximal.

Let F' 3 F be a proper

{i E L / F 15 E F'} is a filter of L / F . Moreover, G # {i}because a E F' - F implies i? E G and i? # i. Also, G is proper because 6 E G would imply 6 = i for some 5 E F' hence 0 A u = x A u for some u E F , therefore 0 = z A u E F', a contradiction.

filter. It is easy to check that G =

Filters, ideals and congruences in lattices l(i)

+ +):

t o check that

Let G be a proper filter o f L / F , G

F’ = {x E L 1 i E G }

tz E ( L / F - G ) shows that x E F +f = t h a t F c F’.

iE

a

4 F’

G. Now t a k e

39

# {i}. It is

easy

is a filter of L . The existence of

i.e. F’ is proper. Then F

E F’

because

L E G, L # i; then b E F‘ - F , showing 0

3.24. Definition. Let PFl(L) (let Spec(L)) denote the set of all prime filters (prime ideals) of the lattice

L.

3.25. Proposition. If L E DO1 t h e n (3.12)

fl {mod F I F E PFl(L)}= A,

.

E L , x # y. Then we have e.g. x $ y. In view of Corollary 3.13 there is F E PFl(L) such that x E F and y # F . Then x A u # y A u for every u E F , otherwise y A u = x A u E F , which would imply the contradiction y E F. Therefore (x,y) $ mod F and this proves the non-trivial Proof. Take x,y

part of (3.12).

0

In the case o f Boolean algebras the above results can be strengthened.

3.26. Proposition. In a Boolean algebra B every congruence p as of the f o r m m o d F f o r

F = {x E B I x p l } . Com me nt .

According t o Definition 5.11, the congruences p of a Boolean algebra satisfy also zpy

+ zpy.

(3.1) and also (3.2) because if x p l and x 5 y then y = z V y p l V y = 1. Thus F is a filter. If xpy then xAypyAy = y and Zpy hence ZAYpy. therefore u = (xAy)V(ZAg)pyVjj = 1. Thus u E F and x A u = x A y = y A u , proving that x mod F y. Conversely, Proof.

Clearly

F

is non-empty, satisfies

40

Lattices, universal algebra and categories

the latter relation means x A u = y A u for some u E because

F ,therefore

xpy

3.27. Definition. Let A be an algebra in the sense of Definition 5.2, c a r d A > 1. Then A is said to be: semisimple if the intersection of all its maximal congruences is the identity AA, and simple if the only congruences of A are the trivial congruences AA and A2. 3.28. Proposition. Every Boolean algebra is semisimple. Proof. From Propositions 3.21, 3.26, 3.18 and 3.25.

0

3.29. Proposition.

T h e following conditions are equivalent f o r a Boolean algebra B : (i) B is simple;

(ii) the only filters of B are (1) and B ;

(iii) B 2 {0,1). Proof.

(i) ($ (ii): From Proposition 3.21, 3.26 and he remark that AB = mod{ 1) and B2= mod B. (ii) + (iii): If a E B - ( 0 , l ) then [ u ) is a proper filter distinct from (1). (iii) +-(ii): Trivial. 0 3.30. Proposition. A filter F of a Boolean algebra B i s maxima2 and only if B / F 2 { & I ) . Proof. By Propositions 3.23 and 3.29.

(OT

equivalently, p r i m e ) if 0

41

Filters, ideals and congruences in lattices

3.31. Remark (cf. lorgulescu (19841). Let m be an infinite cardinal. Following the same idea as in Definition 2.31,

F

it is natural t o define an m-filter as a filter

for every subset S &

satisfying

L such that cards 5 m and A

x exists. A

proper

XES

m-filter (prime m-filter, mazimal m-filter) is defined as a proper filter (prime filter, maximal filter) which is also an m-filter. By an m-prime f i l t e r is meant an m-filter

for every subset S

F such that

CL

such t h a t cards

5m

d

v

XES

x exists. A

m-

mazimal f i l t e r is defined as a maximal element of t h e family of all proper m-filters. A complete f i l t e r (proper complete filter, prime complete filter,

mazimal complete filter, completely-prime filter) is then defined as a filter which is an m-filter (proper m-filter, prime m-filter, maximal m-filter, m-prime filter) for all m. Also, a completely-masimal f i l t e r is defined as a maximal element of t h e family of all proper complete filters. For example every finite filter is complete, every proper principal filter is complete but not

{ X E P(E)Icard(E-X) 5 m} is an m- filter on P ( E ) , while the filter { X E P(E)I X - E is finite} is necessarily prime. If E is an infinite set then

not an m-filter. Clearly every m-prime (completely prime) filter is a prime m-filter (prime complete filter) and every maximal m-filter (maximal complete filter) is an m-maximal (completely maximal) filter. It is conjectured

that the converse implications do not hold in general. The reader is urged t o specify the results of this section t o the corresponding m-concepts.

The following result will be needed in Chapters 7 and 9.

42

Lattices, universal algebra and categories

3.32. Lemma (Rasiowa-Sikorski).

c

Suppose B is a Boolean algebra and let X,, Y, B such that inf X, and sup Y, exist (Vn E I?). Then for every a E B - (0) there is an ultrafilter F of B such that

(i) a E F and for every n E I?: (ii) infX, E F

* z E F (Vz E X,), and

Proof. Let 2, = (jj I y E Y,} ( n E

IN). It follows from

the De Morgan

formulas and Proposition 3.18 t h a t (iii) is equivalent t o property (ii) for 2,. Therefore it suffices t o prove t h e existence of an ultrafilter satisfying (i) and

(ii). First we set a, = inf X, (Vn E I?)and construct by induction a sequence of elements z,

(Vn E $0

E X , such that b, = a A

IN). For

a0

... A (a, V Z), # 0 (a0 V 5 0 ) # 0 for some

V Z0)A

n = 0 the assertion is that a A

E X O :otherwise a A

hence a

(a0

= 0 and a A Z = 0, i.e. a

5 ao, therefore a = a A a.

5 z, for

every z E

Xo,

= 0, a contradiction. The inductive step

is similar, with b, instead of a .

Further we apply Proposition 3.4 to the decreasing sequence X = (€1-1 =

a, bo,bl ,...,b,, ...} : it follows easily that y E [ X ) ++ b, 5 y for some n E IV U (-1) and since all b, # 0 this implies 0 # [ X ) . Thus X is a

[ X )E F , by Corollary 3.14 and Proposition 3.18. Then a E F and for each n E N ,from b, E F we obtain a , V 2 , E F , therefore either a, E F ,in which case X, F , or 0 a, # F ,which implies 2 , E F hence z, # F . proper filter hence there is an ultrafilter F such that

c

43

Monadic and polyadic Boolean dgebras $4. Monadic and polyadic Boolean algebras

T h e monadic Boolean algebras and potyadic Boolean algebras are algebraic structures imposed by t h e predicate calculus (see Halmos [1962]). In this section we shall present in an outlined form some basic results on these structures. See also Daigneault [1967].

4.1. Definition. A n ezistential quantifier on a Boolean algebra such that, for any p , q E

A , the following

(4.1)

30 = 0 ;

(4.2)

PI3P;

(4.3)

3 ( p A 3q) = 3 p A 3q .

A

is a function 3 :

A +A

properties hold:

A monadic Boolean algebra or m o n a d i c algebra for short is a pair ( A , 3 ) , where A is a Boolean algebra and 3 is an existential quantifier o n A . 4.2. Definition. Let ( A , 3 ) , (A’,3) be two monadic algebras. A m o r p h i s m of m o n a d i c algebras f : ( A , 3 )+ (A’,3)is a Boolean morphism f : A t A‘ such t h a t

f ( 3 p ) = 3 f ( p ) , for any p E A . 4.3. Example.

B be a Boolean algebra and X a non-empty set. Suppose A is a Boolean B X such that, for any p E A , there exists i n B t h e following supremum: V { p ( t ) It E X } . Suppose A is closed w i t h respect t o the Let

subalgebra of

operation 3 defined by

(3p)(y) = V { p ( t ) I t E X }

,

for any p E A and y E X

Then one can prove t h a t 3 is an existential quantifier on called a B-valued functional m o n a d i c algebra.

A . ( A , 3 ) will

be

Lattices, universal algebra and categories

44

4.4. Definition. A u n i v e m a l quantifier on a Boolean algebra A is a function V : A -+ A such that, for any p , q E A , we have:

(4.4)

v1 =1 ;

(4.5)

VPSp ;

(4.6)

V ( pV V q ) = V p V V q

4.5. Lemma. In a m o n a d i c algebra ( A , 3 ) t h e following properties are t r u e : (4.7)

31=1;

(4.8)

33=3;

(4.9)

p E 3(A) H 3 p = p ;

(4.10)

P5 9

(4.11)

3(3p)=3p;

(4.12)

3 ( A ) is a Boolean subalgebra of A ;

(4.13)

3(Pvq)=3Pv39;

(4.14)

3P

+

+- 3P 5 39 ;

S 3(p +9)

-

4.6.Remark.

In a monadic algebra ( A ,3) we can define a universal quantifier V by V p = 3,

for any p E A . It is obvious that a monadic algebra can be also defined using a universal quantifier.

4.7. Definition. A m o n a d i c subalgebra of a monadic algebra ( A ,3) is a Boolean subalgebra A’ of A which is closed under 3. A m o n a d i c ideal of ( A , 3 ) is an ideal M

M . By (4.14), the quotient Boolean algebra AIM has a canonical structure of monadic of the Boolean algebra A such that p E M implies 3 p E

Monadic and polyadic Boolean algebras

45

algebra and t he natural Boolean morphism h : A + A / M is a morphism of monadic algebras. 4.8. Proposition.

Let ( A , 3 ) be a monadic algebra and B = 3 ( A ) . FOT a n y ideal M of the Boolean algebra B we denote by M the Boolean ideal of A generated by M . Then: monadic ideal of A ;

(a)

M is

(b)

M nB

(c)

If N is a monadic ideal of A, then N r l B = N ;

a

= M ; M = 3-'(M);

(d) T h e m a p M H M gives a n isotone bijection between the ideals of B and the monadic ideals of A; (e)

If M is a m a x i m a l ideal of B , t h e n M is a m a x i m a l monadic ideal of A.

4.9. Definition.

A monadic algebra ( A , 3 ) is simple if (0) is the unique monadic ideal of A; ( A , 3 ) is semisimple if the intersection of i t s monadic ideals is (0). An existential quantifier 3 is simple if p # 0 implies 3 p = 1. One can prove that a monadic algebra ( A , 3 ) is simple iff t h e quantifier

3 is simple. 4.10. Proposition.

(i) A n y monadic algebra is semisimple. (ii) A n y monadic algebra is isomorphic t o a subdirect product of simple monadic algebras. Proof.

(i) By Proposition 4.8 and th e semisimplicity of Boolean algebras.

Lattices, universal algebra and categories

46 (ii) Straightforward.

0

4.11. Definition.

A constant of a monadic algebra ( A ,3) is a Boolean morphism c : A + A such t h a t c3 = 3 and 3c = c . A monadic algebra ( A , 3 ) is rich if for any p E A there exists a constant c of A such that 3 p = c ( p ) . For any constant c and every monadic ideal M of ( A , 3 ) the following properties hold: (4.15)

p E ](A)

(4.16)

c ( A ) = 3 ( A ) ; cc = C ; CV = V; VC= c ;

(4.17)

p E M implies c ( p ) E M ;

(4.18)

there is a unique constant C of

c(p) =p ;

A / M such that hc = Ch .

4.12. Lemma.

A n y rich monadic algebra ( A , 3 ) is isomorphic t o a functional monadic algebra. Proof. Let B = 3 ( A ) and map f : A -+

X th e

set of constant of

( A , 3 ) . Consider the

BX defined by

f ( p ) ( c )= c ( p )

,

for any p E A and c E X

( A ,3) and th e properties of constants one can show that f ( A ) is a B-valued functional monadic algebra and f is an isomorphism Using the richness of

between A and f(A).

0

4.13. Lemma.

A n y simple monadic algebra ( A , 3 ) is rich. Proof. If p E A, p

#

0, then there exists a Boolean morphism c : A -+ {0,1} such that c ( p ) = 1; c can be considered a Boolean morphism c : A -+ A such that c(A) = {O,l}.

Monadic and polyadic Boolean algebras

47

Since 3 is simple, we can show that c is a constant of

A such t h a t

3 ( p ) = c ( p ) = 1. 4.14. Lemma. A n y direct product of rich m o n a d i c algebras i s a r i c h m o n a d i c algebra. 4.15. Theorem (Halmos). A n y m o n a d i c algebra is isomorphic t o a f u n c t i o n a l m o n a d i c algebra. Proof (given by Leblanc [1967]). By 4.10(ii), 4.13, 4.14 and 4.12.

0

4.16. Def inition. A polyadic (Boolean) algebra is a structure ( A ,U,S, 3), where A is a Boolean algebra,

U

a non-empty set, S is a function from

endomorphisms of

Uu to

t h e set of

A and 3 is a function from P(U)t o t h e set of existential

quantifiers of A such that the following axioms hold:

(4.19)

S(lu) = 1~;

(4.20)

S(OT)= S(O)S(T),

(4.21)

3(0) = 1~;

(4.22)

3 ( K U K ' ) = 3(K)3(A7'),

(4.23)

S ( a ) 3 ( K )= S ( ~ ) 3 ( 1 <,) such t h a t

(4.24)

~

o ~ U -=~

for any

1

E 'U

for any

;

~

,

;

K,K'C U

for any I< C

3 ( K ) S ( a )= S(o)3(0-'(1{)) such that oI.-I(K)

O,T

;

U and O , T

E

Uu

~ for any K

U and

a E Uu

is injective.

The notion of m o r p h i s m of polyadic algebras or polyadic m o r p h i s m for short is defined in a natural way. Card(U) will be called the degree of the polyadic algebra ( A ,U,S, 3). A subset I< o f U will be called a support of an element p of A if 3(U - IC)p = p ; p is i n d e p e n d e n t of K if 3 ( K ) p = p . A polyadic algebra is locally f i n i t e if every element has a finite support.

Lattices, universal algebra and categories

48 4.17. Lemma.

FOTany polyadic algebra ( A ,U,S, 3) such that Card(U) 2 2, the following properties are equivalent:

(i) K is a support of p ;

All the polyadic algebras considered i n the sequel are supposed to be locally finite and of infinite degree. 4.18. Remark.

( A ' , U , S , 3 ) be two polyadic algebras and f : A t A' be a Boolean morphism such t h a t S(a)f = f S ( u ) ,for any u E U u . Then f is a polyadic morphism iff f ( 3 ( i ) p ) = 3 ( i ) f ( p ) ,for any p E A and i E U. Let ( A , U ,S,3),

4.19. Example. Let B be a complete Boolean algebra, set.

U

an infinite set and X a non-empty

Denote by Set(Xu,B) the set of all functions

Xu

t

B . For any

IC 5 U and x,y E Xu put xK*y w

If IC

U

and

X1U-K

= Y1U-K

.

E 'U define

3 ( ~ ): Set(Xu,B ) t Set(Xu,B )

S(U)

:

Set(Xu,B ) + Set(Xu7B )

X u + B , u E'U and K E U. It is straightforward to show t h a t Set(Xu, B ) becomes a polyadic algebra.

for any p :

49

M o n a d i c a n d p o l y a d i c Boolean algebras 4.20. Lemmq. F o r a n y p : X u + B and K

s U t h e following properties are equivalent:

(i) K i s a support of p in Set(Xu, B ) ;

W e shall denote by F ( X U ,B ) the set of the elements of ving a finite support. lt is obvious that

S e t ( X u , B ) ha-

F ( X U ,B ) is a locally finite polyadic

algebra. 4.21. Definition.

A polyadic subalgebra of F ( X U ,B ) will be called B-valued f u n c t i o n a l polyadic algebra. 4.22. Lemma.

In a polyadic algebra ( A , U,S, 3 ) t h e following properties hold: (i)

If p E A, t h e n {I< [ p i s i n d e p e n d e n t of I<} is a n ideal of P(U)a n d the set {I< I I< i s a support o f p } is a filter of P(U);

(ii) I f p is independent of K a n d K' C U , t h e n 3 ( K ' ) p = 3(K' - I+;

(iii) If I<' is a support o f p a n d K E U ,t h e n 3 ( K ) p = 3 ( K n Ir'')p a n d K' - I( is a support of 3 ( K ) p . 4.23. Lemma.

Let ( A ,U, S, 3 ) be a polyadic algebra, p E A; u,r E U u a n d I<, Is7' such that (i)

K i s a f i n i t e support

(ii)

ulKnKr

ofp;

is injective;

(iii) u ( K n K ' ) n r(IC - K ' ) = 0;

U

50

Lattices, universal algebra and categories T h e n S(r)3(IC')p= 3(u(IP n K))S(a)p.

4.24. Proposition. L e t ( A ,U , S, 3 ) be a polyadic algebra. F o r a n y p E A, r E U u and K' C U w e have t h e equality: (4.27)

S(r)3(K')p= V {S(a)pI ~ I { : T }.

Proof. It is easy to see that S(r)3(K')p2 S(a)p,for any uI<:r. Consider q E A such that q 2 S(o)p, for any uI
(j/i)(k) =

j ,

ifk=i

k,

ifkfi.

4.25. Corollary. In a n y polyadic algebra ( A ,U , S , 3 ) w e have: (4.28)

3(i)p = V {S(j/i)p 1 j E U } .

4.26. Theorem (Halmos [1962]). A n y polyadic algebra ( A ,U , S, 3 ) is i s o m o r p h i c t o a f u n c t i o n a l polyadic algebra. Proof. Let B be the MacNeille completion of the Boolean algebra A . Consider the map

defined by

@ ( p ) ( a )= S(a)p ,

for any p E A and u E U'

Monadic and polyadic Boolean algebras Using Proposition 4.24 we can show that

51 is an injective polyadic mor-

phisrn.

0

4.27. Remark. Let ( A ,U,S, 3) be a polyadic algebra. For any p E A we shall denote by K P the minimal support of p . Consider p E A and let of

U

such that

KP C U’. For any r E UtU and I<’

U’ be a countable subset

U

we have a modified

form of (4.28):

(4.29)

S(r)3(K‘)p= V { S ( a ) p1 0 E U”, aI<;r} .

The following representation theorem is t h e main result of this section. 4.28. Theorem (Halmos 119621).

Let ( A ,u,s, 3) be a polyadic algebra and M a proper filter of the Boolean algebra

T h e n there exists a n o n - e m p t y s e t X and a polyadic m o r p h i s m @ : A F ( X U , L z )s u c h that @ ( p ) = 1, f o r a n y p E M . Proof (Potthoff [1971]). Let any

Sw(A)be the set of finite subsets of A and for

F E Sw(A)we denote

k = {F‘ E Sw(A)I F Consider an ultrafilter

{ kI F

D

F’}

.

of the Boolean algebra

E Sw(A)}C

P ( S w ( A ) )such that

D.

F E Sw(A),set ICF = U {KP I p E F } and choose a countable p E F and IT f Sw(U) subset UF of U such that KF C UF. If F E Sw(A), For any

then we can prove that the set

( ( S ( 0 ) p 1 0 E u;, aK*r}17- E is countable. Since

F

u:}

is finite, the following set is countable:

Lattices, universal algebra and categories

52

Applying t he Rasiowa-Sikorski Lemma 3.32 we get, for any F E Sw(A),an ultrafilter N F of A having the following properties: (4.30)

F n M E NF ;

(4.31)

for any p f A , (T E

UF

and

K

c Iir~we have:

S ( a ) 3 ( K ) pE NF H there is T E UF, such that TIC*(Tand S ( r ) p E

II

Consider t he ultraproduct

NF .

U p / D (cf. Definition 5.46) and for any

FESu(A)

rE

( II

UF)'

define

F

by putting

( r / D ) ( i )= r ( i ) / D and r ~ ( i=) r ( i ) ( F )

)

Define the map @ : A

for any p E A and r E

( II F

--$

for any i E U

.

F ( ( II V,/D>")Lz) by: F

UF)'. Using the properties of t h e ultraproduct

one can prove t h a t i9 is well-defined. It is straightforward t o show that i9 is

Uu. Now we shall prove that 3(IC)@(p)= @ ( 3 ( K ) p ) ,for any p E A and K U. It is enough to consider the case IC C Kp. It is easy t o see that {F I K 5 K F } E D ,therefore we have: a Boolean morphism which commutes with S(a),for any

(T

E

Monadic and polyadic Boolean algebras

53

Q ( 3 ( K ) p ) ( r / D= ) 1 ++ {F I S ( r F ) 3 ( K ) pE

NF}

E

D

H

++ {F I S ( r ~ ) 3 ( K )EpN F , K G K F } E D

I

By (4.31), t he latter condition is equivalent t o (4.33)

{F I there is a~ E Ug, S ( a ~ )Ep N F , a ~ l ( t r K ~ , ICF}ED .

Using again (4.34)

{F I K G K F } E D ,(4.33)

is equivalent t o

{ F 1 there is OF E V g , S ( a ~ )Ep N F , O F K J F }E D

.

But

3 ( W Q ( p ) ( r / D )= v { @ W ( s / D ) I ( s / D ) K * ( r / D ) } therefore 3(11)Q(p)(r/D)= 1 is equivalent t o (4.35)

there is ( s / D ) K ( r / D ) ,such that { F I S ( s ~ ) Ep N F } E D

.

That (4.34) is equivalent t o (4.35) is straightforward, therefore @ is a polyadic morphism. The rest of the proof is obvious.

0

4.29. Remark.

This representation theorem is th e algebraic counterpart o f the completeness theorem o f predicate calculus.

Lattices, universal algebra and categories

54 $5. Universal algebra

In this section we sketch those ,dndamental concepts of universal algebra t h a t will be used in t h e remainder of this book. For a complete introduction see e.g.

Cohn [1965] or Pierce [1968]; for an encyclopedic treatment the

reader is referred t o Gratzer [1968]; cf.

also Burris and Sankappanavar

[1981]. T h e basic concepts and results presented below are abstractions o f similar concepts and results encountered i n various “concrete” fields of algebra such as group theory, module theory, lattice theory etc.

5.1. Definition. If A is a set, n E nV and f : A” + A , we say t h a t f is an n - a r y operation on A . We also say t h a t n is the arity o f f . It is convenient t o distinguish between constant operations of a r i t y n (i.e. f(xl, ...,s,) = a for some a

>0

E A and all x l , ...,x, E A ) and operations of a r i t y zero (i.e. f : A’ = 0 + A or simply f E A w i t h no arguments).

5.2. Defin ition. A type of algebras is a map r : A + nV where A is a non-empty set. A n algebra of type r or simply a r-algebra is a pair

A is a set and each f~ is an operation of arity .(A) o n A . W e refer to fx(A E A) as the basic operations of the algebra. T h e same notation A is often used for the underlying set A of A and t h e algebra A itself. If r is a f i n i t e type, i.e. t h e set A is finite, say A = {l,..,m}, then r can be

where

represented by the vector ( r ( l ) ..., , ~(rn)). T h u s e.g. a lattice (L, A, V) is an algebra of type (2,2), a group (G, -,- l , e )

is an algebra of type (2,1,0) etc.

55

Universal algebra

5.3. Definition. A subalgebru of an algebra ( A , { ~ A } A ~ isA )a subset S

A that is closed

with respect to the basic operations, i.e. (5.2)

51,**.,ZT(A) E

s * fA(Z1,

..a,

&(A))

E

s (vx E A)

5.4. Remark.

A subalgebra S of a 7-algebra (5.1) is itself a 7-algebra with respect t o the restrictions of th e basic operations fA of A t o t h e elements of S. Note for example that a subsemilattice of a semilattice

( L ,A) (6. Defi-

nition 1.33) is a subalgebra of ( L , A ) in the sense of Definition 5.3; similar remarks hold for the concepts of sublattice, 0-1-sublattice, De Morgan subalgebra, Boolean subalgebra and monadic subalgebra introduced in Definitions 1.33, 2.24 and 4.7.See also Remark 1.34 and Example 2.25. 5.5. Definition.

X of an algebra A we denote by X t h e intersection of all subalgebras of A that include X and we call X the subalgebra generated b y X. For every subset

The justification of the above definition originates in the obvious fact that the subalgebras of an algebra A form a Moore family of the complete lattice P ( A ) ;cf. Example 1.13 and Definition 1.14. In view of Remark 1.15, the map which sends every subset X C A to

X

is a closure operator and X

is the least subalgebra including X . We give below an intrinsic characteri-

zation of the elements of

R.

5.6. Definition.

Let ( A , { ~ A } A ~ Abe ) a .r-algebra, X C A and a E A . By a f o r m a t i v e cons t r u c t i o n of length n , of the element a from the elements of X w e mean a

finite sequence a l , ...,a , of elements of A such that a, = a and for each

i

E (1, ...,n } one of the following situations hold:

(a) ai

E X, or

56

Lattices, universal algebra and categories

(b) there exist X E A and

kl,

...,k , ( ~ < ) i such that

ai = fx(xkl,

...,~ k , ( ~ ) ) .

5.7. Proposition. The subalgebra X generated b y a subset X of an algebra A coincides with the set of all elements of A having a formative construction from the elements of X .

Y be the set of all elements having a formative construction. Then X s Y because every x E X is a formative construction o f length 1. The set Y is a subalgebra of A because, for each X E A, if q ,...,xc,(x)E Y Proof. Let

then we can concatenate the formative constructions of these elements and

add a t the end the element y = fx(xl, ...,x.(~)) obtaining in this way a new formative construction, therefore y E that

X

S, then

s

Y

Y. Finally if S

is a subalgebra such

S follows easily by induction on t h e length of a

formative construction of an arbitrary element y E

Y.

0

5.8. Corollary (Definitions by Algebraic Induction). A subset S A coincides with the subalgebra X generated b y X if and only i f the following conditions hold:

s

(i)

if x E X then x E S;

(ii) if

...,z,(x) E S then fx(z1, ...,z,p)) E S (VX E A);

z1,

(iii) every element of S is obtained b y applying rules (i) and (ii)finitely many times. Proof. This is a paraphrase of Proposition 5.7: rules (i) and (ii) state that

the elements of the formative constructions belong to S while rule (iii) says t h a t every element of S has a formative construction.

5.9. Corollary (The Compactness Property). For each element z E X there is a finite non-empty subset

X such that

...,zn}.

z E (21,

0

...,x,}

(51,

57

Universal algebra Comment.

We mark this fact by writing z = ~ ( 2 1..., , 2,).

Proof. Let 2 1 , ..., z, E X

be the elements occurring in a formative construc-

tion of z .

0

Another corollary establishes a method for proving properties of elements in an algebra. The term "property" can be understood either intuitively, or according t o the following formal definition: a property defined on a set simply a map

:

7r

A

A

is

-+ {0,1};for every a E A , the property is interpreted is 1or 0.

intuitively as being true or false according as .(a) 5.10. Corollary (Proofs by Algebraic Induction).

Let X be a subset of an algebra ( A , {fx}xE+,).

(i)

?r

is true f o r all

2

7r

7r

satisfies

E X , and

(ii) (VX E A) (Vzl, ...,".(A) E A ) if fx(z1, ...,z.(q) satisfies ?r; then

If a property

$1,

...,x.(x)

satisfy

7r

then

i s true f o r all z E X ,

Proof. The set

Y

of all elements of A satisfying

a subalgebra by (ii),therefore X

7r

includes X by (i) and i s

CY.

0

5.11. Defi nitio n.

A congruence on a .r-algebra (A,{fx}xE~) is an equivalence relation

A such that

-

-

on

E A. The quotient algebra of the algebra A by the congruence is the pair (A, { j x } ~ , = ~ ) ,where A = A / and fx : AT(x)--+ a (A E A)

for every X

are defined by

N

Lattices, universal algebra and categories

58

5.12. Proposition. The quotient algebra of a r-algebra is a r-algebra. Proof.

For every X E

z1 N

and

...

and

-

A, if 21 = 2; and ... and ST(x) = then x : ( ~ )therefore , (5.3) implies fx(zl, ...,z,(x)) N

fx(zi,...)x : ( ~ , ) i.e. , the operations fA are well-defined.

0

T he construction of th e quotient algebra is well known i n each "con-

H is a normal subgroup of a group G t h e construction of t h e quotient group G / H obeys t h e general scheme of Definition 5.11, taking as th e congruence associated w i t h H , i.e. z y iff H z = Hy. crete" field of algebra. Note e.g. th a t if

-

N

5.13. Remark.

A congruence relation on t h e algebra A is a subalgebra of t h e direct product A x A. (The direct product o f a family {(Ai,{ f i x } x E ~ ) } i E ~ of r-algebras is the r-algebra II A ; , { f x } x E A ) , where for every X E A and every

(

$1

5.14. Definition. Let

A = ( A , { f x } x E ~ ) and 0 = ( B ,{ g X } X E A ) be r-algebras. A

T - ~ O ~ O ~ O T -

phism (also called a 7-morphism or simply a homomorphism or a morphism when no confusion is possible) from A to B is a map 'p : A + B which commutes with the operations f x , i.e.

A + B designates t h e fact t h a t 'p is a homomorphism, while Hom(A, B) or simply Hom(A,B ) denotes t h e set of

for every X E A. The notation

'p :

59

Universal algebra

all homomorphisms from A t o B. If one works simultaneously with various types of algebras it is convenient t o provide Hom with an index designating the type. The term 7-isomorphism designates a 7-homomorphism which is also a bijection. We say that two 7-algebras A and B are isomorphic

A M 0 provided there is a 7-isomorphism between them. The 7-homomorphisms and 7-isomorphisms from A t o A are also referred t o as the (T-) endomorphisms and (T-) automorphisms of A, respectively. and we write

The various concepts introduced in Definitions 1.20, 1.21, 1.27, 2.19 and 2.22 are particular cases of the universal-algebra concepts in Definition 5.14. Also, it will be tacitly assumed that the homomorphisms (isomorphisms, endomorphisms, automorphisms) corresponding t o the various classes of algebras to be studied in this book are given by Definition 5.14 applied t o the types of those algebras. 5.15. Remark. Several well-known properties of homomorphisms are easily transferred from various “concrete” fields of algebra t o the above general concepts. Thus e.g. the identity map 1~ is an automorphism. The composite of two homomorphisms (isomorphisms) is a homomorphism (an isomorphism). The relation

M

is reflexive, symmetric and transitive. If cp

E Hom(A,B), S is a subalgebra of A and T a subalgebra of B , then cp(S) is a subalgebra of B and cp-l(T) is a subalgebra of A . If is a congruence of A and A = A / -, the canonical map^ : A + A is a homomorphism.

-

5.16. Proposition (The Homomorphism Theorem).

Let cp : A + B be a homomorphism. T h e n : (i) (5.6)

T h e relation kerA,B cp

x ker cpx’

OT

simply ker 9,defined by

cp(x) = cp(z‘)

i s a congruence o n A .

(ii) FOT every algebra

C and every homomorphism II,

:

A

+

C such

Lattices, universal algebra and categories

60

that ker cp ker II,there is a unique homomorphism x : A / ker cp C such that II,= x o *.

(6) The map

x

:

---t

A / ker cp -+ cp(A) defined b y

(5.7)

is an isomorphism. Proof. Essentially th e same as i n each “concrete” field o f algebra.

0

5.17. Corollary.

If cp

:

A

t

B is

a

surjective homomorphism then B

M

A / ker cp.

Next we turn to classes of algebras and free algebras. T h e t e r m class will be given t h e usual meaning in set theory. 5.18. Defi nitio n . Let IC be a class of r-algebras and {Ai}iEIa family o f IC-algebras, i.e. A; E

K

(Vi E I ) . A subdirect product in K: of these algebras is a subalgebra S of the direct product ( I I A i ) E K such th a t S E IC and for every i E I and every xi E Ai there is x E S such that 1~iz= x i , where 7ri : IIAi -t A; are the canonical projections. A subdirect (direct) decomposition in K of a K-algebra A is an isomorphism between A and a subdirect product S (the direct product) of a family { A i } i E Io f K-algebras. W e usually express a subdirect (direct) decomposition i n th e form f : A + II A;, where f = L o g , g :

A

(where

--t

f

S is th e isomorphism and is t h e isomorphism).

L

: S --f IIAi is t h e inclusion mapping

T h e decomposition is said to be proper if

no factor Ai is isomorphic to A . T h e algebra A is said to be subdirectly

K: if cardA > decomposition in K.

(directly) irreducible in (direct)

1 and A has n o proper subdirect

5.19. Theorem (Birkhoff). Every r-algebra has a subdirect decomposition into subdirectly irreducible factors (in t he class of all r-algebras).

61

Universal algebra Proof. See e.g. Birkhoff

0

(19671or Pierce [1968].

5.20. Definition.

X , we denote by X * the set of all finite sequences of elements of X , including the "empty sequence" A. We then refer t o X as an alphabet, i t s elements are called letters and those of X' are termed words. If 2,,..., xm E X , the word (21,..., xm) will be denoted simply by For every non-empty set

concatenation: zl...x,, and said t o be a word of length m. The words ~ ~ . . . x ~ +(k~ = z k1, ...,m - 1) are called the segments of 21. .AT,.

More ge-

nerally, given the words wl,...,wn, we denote by wl...wn the word obtained by their concatenation.

5.2 1. Definition. Let X be a non-empty set.

F n X' = 0. (W,{Fx}AEA), where for

such t h a t

Take a A-indexed set

The set W =

(X

F = {FA1 X

E

A)

U F)' is made into a T-algebra

E A the operation denoted FA maps every sequence (wl, ...,w,(l)) E W'@)t o the word FAw ,...w,(X) E W . By the Peano (T-) algebra on the set X of free generators we mean the subalgebra P ( X ) of (W,{ F A } A ~generated A) by X . each X

We will prove that the Peano algebra is a free algebra in the following sense.

5.22. Definition. Let: R be a class of T-algebras, A a r-algebra and X a set. Then A is said t o be the free algebra i n the class R (or simply t h e free R - a l g e b r a ) on the set X of free generators (or equivalently, freely generated by X ) provided the following conditions hold: (a) A E

R;

(b) there is an injection i (b,)

i ( X ) = A and

:

X +A

such that

Lattices, universal dgebra and categories

62 (b2) for every algebra B. E

R

unique homomorphism (p : A

If-as

is usually th e case-X

i ( z )= z (Vz) then (p

o

B

and every map cp : X -+ t

5 A

B

such t h a t (p o

there is a

i = cp.

and i is the inclusion mapping

i = cp reads (p I X = cp.

5.23. Lemma.

If w E P ( x ) there is no segment of w belonging to P ( x ) . Proof by induction of th e length n of w. If n = 1 (i.e. w E X or with .(A)

w = FA

= 0) then w has no segment. For the passage from n t o n + l take

w E P ( X ) of length n + l >_ 2. Inasmuch as P ( X ) is generated by X , w is of

w = FAwl...w,(~) for some X E A and wl, ...,w,(A) E P ( X ) where r ( X ) 2 2. If w' is a segment of w and w' E P ( X ) then w' = FAW; ...w:(~) for some wi,,..,w:(~) E P ( X ) . This implies there is an index k E (1, ...,.(A)} the form

such that wI, = segment of

Wk ,

Wh

( h = 1,...,k

- 1) and

WE, # wk.

Therefore

which contradicts the inductive hypothesis.

5.24. Lemma.

WE,

is a 0

x

FOT every w E P ( x ) either w = x E or w is uniquely represented in the f o r m w = F A W...~w,(A) with w1,...,w,(x) E P ( X ) . Proof.

For n = 1 the statement is trivial.

For n

>

1 suppose there

is w E P(X) of length n and having two distinct representations w =

FAW ...w,(x) ~ = Fpw~...w~(,~ with A, p E A and w1, ...,w,(A), wi, ...,w&) E P ( X ) . It follows that FA= F, therefore X = p and w1...w,(x) = wi ...w', ( P I . This implies further th e existence of an index k E {1,..., r(X)}such t h a t w; = wh ( h = 1,..., k - 1) and w6 # Wk. Therefore wi i s a segment of W k , which contradicts Lemma 5.23.

0

5.25. Theorem.

FOT every non-empty set X , the Peano r-algebra P ( X ) on X is the free algebra freely generated b y X in the class of all r-algebras. Proof. Conditions (a) and (bl) in Definition 5.22 are fulfilled for the inclusion mapping

i

: X

--f

P ( X ) . To prove (b2) let ( A , { ~ A } x ~be A )a 7-algebra

Universal algebra

63

and y : X + A. Define a relation (p

c P(X) x A

by algebraic induction

in the algebra P(X) x A:

(a) ( x , y ( z ) ) E p for every z E X,

(7)every element ( w ,y) E (p is obtained by (a)and (p). In view of Lemma 5.24 every w E P ( X ) appears as the first component of an element ( w , y ) E (p exactly once, therefore (p is a function

A . Now conditions (a)and (p) read F I X = y and (p(Fxwl...w,(x)) = fx ((p(wl), ...,(p(wT(x))), i.e. (p is a homomorphism. Finally if 1c, E H o m ( P ( X ) , A ) and 1c, ] X = cp it is easily proved by algebraic 0 induction in P ( X ) that $(w) = (p(w) for every w E P ( X ) . (p

: P(X)

---t

Our next step is t o introduce equational classes and their free algebras. The basic concept is that of identity o f an algebra. To understand Definition 5.26 below take the simple example of the identity

of a semigroup

(G,.). Let (P(X),F) be t h e

Peano algebra of type (2) and

take wl,wz E P(X) defined by w ~ ( ~ , x ~ = ,F zF x~1 )x 2 x 3 and

w2(x1, ~

2 ~ x = 3 )

Fq

F5223,

respectively (cf. the comment t o Corollary

5.9). Then for every cp E H o m ( P ( X ) , G ) such that c p ( z k ) = a k (k = 1,2,3) we have y(wl)= (al . u 2 ) a3 = al - (a2 . a3) = cp(w2), so t h a t (5.8) can be viewed as expressing the property cp(wl) = y(w2) (Vcp E Hom(P(X), G)). 5.26. Definition. Let P(X) be a Peano r-algebra and set

(5.9)

I d A = n {ker y I y E H o m ( P ( X ) , A)} ,

(5.10)

IdK = n {IdA I A E K )

Lattices, universal algebra and categories

64

for every r-algebra A and every class K of r-algebras. If (wl, w2) E I d A we say t h a t (w1,w2) is an identity of A or the identity w1 = w2 holds in A;

the pairs in IdK are called the identities of the class

K.

5.27. Proposition.

Let P(X) be a Peano r-algebra and

K

a class of r-algebras. T h e n

(i) P(X)/IdK: is a r-algebra and

(ii) f o r every algebra A E K and every m a p v : X --f A there is a unique h o m o m o r p h i s m V E Hom(P(X)/Id K ,A) saeh that v = V 0-0i, where i : X + P(X) i s the inclusion m a p a n d ^ : P ( X ) + P(X)/IdK i s the canonical surjection. Proof.

(i) IdK: is a congruence in view of Proposition 5.16(i) and the easy remark that the congruences of an algebra form a Moore family. Therefore P(X)/IdK is a r-algebra by Proposition 5.12.

i

Fig. 5.1.

(ii) In view of Theorem 5.25 there is a unique cp E Hom(P(X),A), such that v = cp o i. Inasmuch as ker^= IdK kercp by (5.9) and (5.10), Proposition 5.16(ii) (applied for A := P(X), B = C := P(X)/IdK:, cp := *, C := A, 1c, := cp) yields the existence of a unique V E Hom(P(X)/IdK:,A) such that cp = V o It follows that v =

i = V 0 - 0 i. To prove the uniqueness of fi with the latter property, take 1c, E H o m ( P ( X ) / I d K , A ) such that v = 11, 0 - 0 i. Then 11, o A = cp by the uniqueness of cp, therefore 11, = E by the uniqueness of V. 0 cp o

65

Universal algebra 5.28. Definition.

A class

K

of .r-algebras is called a variety or an equational class provided

A x K C_ P ( X ) x P ( X ) such that

there is a set

A E K e A x K 2 IdA

(5.11)

for any r-algebra A (cf. Definition 5.26). Many important classes of algebras, e.g. semigroups, groups viewed as algebras of type (2,1,0), lattices, Boolean algebras a.s.o., are defined by systems of identities; such a system of axioms can be viewed as establishing an equivalence of the form (5.11) so that the corresponding classes of algebras are equational in the sense of Definition 5.28. 5.29. Lemma.

T h e following conditions are equivalent f o r a class

(i)

K

K

of r-algebras:

is a n equational class;

IdK G Id A) f o r a n y .r-algebra A;

(ii) (A E K

ij

(iii) (IdK

IdA 3 A E K ) f o r a n y r-algebra A.

Proof.

(i) + (iii). It follows from (i) that A x K G

I d R C IdA

(l IdA = IdK therefore AER

+ A x K G IdA + A E K.

(iii) =$ (ii). Because th e implication converse t o (iii) is trivial. (ii) =+ (i). Trivial. It is easy t o see that if

K

is a variety and

0

A E K then every subalgebra

of A is in K ,every homomorphic image of A (or equivalently, every quotient algebra of A) is in

K

and every direct product of algebras of

K

is in

K ;cf.

Definitions 5.3, 5.11, 5.13, 5.14 and Corollary 5.17. (Remark 1.1.34 and the remark following Definition 1.2.24 are particular cases of the above remark.)

The converse also holds, i.e. a class K of r-algebras is equational if and only

Lattices, universal algebra and categories

66

if it is closed with respect t o taking subalgebras, homomorphic images and direct products. This is the Birlcho$ T h e o r e m which we quote here without proof.

5.30. Theorem. Let K be a n equational class of r-algebras and P(X) the Peano r-algebra

o n X. T h e n P(X)/IdK is the free K-algebra o n the set X. Proof.

let i

: X + P(X) be the inclusion map and

: P(X) -+

A

P(X)/IdK the canonical surjection. Then- o i : X -+- P(X)/IdR is an injection. From P(X) = X we obtain P(X)/IdK = X = o i)(X), i.e. condition (b,) from Definition 5.22, while (b,) holds by Proposition 5.27. It remains t o prove (a), i.e. P(X)/IdK E K. In view of Lemma 5.26 this is equivalent t o IdR Id(P(X)/IdR). Thus we take (w1,wZ) E IdK and according t o Definition 5.26 we must prove that cp(w1) = cp(w2) for every cp E H o m ( P ( X ) , P ( X ) / I d K ) . Set (p(wk) = ziji ( k = 1,2). Further we t a k e A E K: and 1c, E H o m ( P ( X ) , A ) ; then ker^ = IdK I d A hence by Proposition 5.16(ii) there exists x E H o m ( P ( X ) / I d K , A ) such t h a t 1c, = x o inasmuch as x o cp E H o m ( P ( X ) , A ) it followsthat IdK ker(Xocp) hence(wl,w2) E ker(Xocp)

C

c

c

A.

and

(wi,wi) E ker4 for every 1c, E H o m ( P ( X ) , A ) and every A E K , therefore (wi,w:) E IdR or equivalently ziji = G;, that is cp(wl) = cp(w2). Thus 0

The remainder of this section is based on the following remark. Roughly speaking, t h e object of universal algebra is the study of the general concept of algebra, while other fields of algebra deal with “more concrete” algebraic structures, like groups, rings, modules, lattices etc. Note however that there may be several ways of thinking of a “classical” algebraic structure as a .r-algebra; thus e.g. a group can be viewed not only as an algebra of type

67

Universal algebra

(2.1,O) but also as an algebra (G,.) of type (2). We are going t o concentrate a bit more on this point.

5.31. Definition.

Let (A,{f,}xE~) be a 7-algebra and X a non-empty set. The set AAX=

{f 1 f

: AX

4

A} is made into a 7-algebra ( A A X{f,,},,~) ,

X E A and every

01,

where for every

...,o , ( ~E) AAX,fx(ol,...,o,(,)) is defined by

we usually write simply

fx instead of

f,,.

5.32. Definition. Let A be a 7-algebra and

X

a non-empty set. By A[X] we denote the

subalgebra of AAX generated by the set (5.13)

T ~ ( V= ) W(Z)

{7r,},E~

of projections, defined by

(VW E Ax)

for each z E X. The elements of A [ X ] are said t o be t h e polynomials of A in the variables from X. An alternative description of polynomials is given in Definition 5.33 jus-

tified by Proposition 5.34. 5.33. Definition. Let P(X)be a Peano 7-algebra and A a 7-algebra. The canonical m o r p h h m N

: P(X) 4 AAXis the homomorphic extension of the map T : X

defined by ~ ( x=) 7rz

AAX (Vx E X). For each w E P(X), 6 is called t h e poly-

nomial generated by w. 5.34. Proposition.

P p )= A[X]. Proof. By double inclusion, each of them by algebraic induction:

4

Lattices, universal algebra and categories

68

5.35. Proposition.

Homomorphisms commute with polynomials (2.e. if cp E Hom(A, B ) and w E P ( X ) then ~ ( 6 =)

cpF)).

Proof. Algebraic induction. The canonical morphism

0

N

yields an alternative description of t h e iden-

tities o f an algebra.

5.36. Lemma.

Let P ( X ) be a Peano .r-aEgebra, A a r-algebra, V E Hom(P(X),A) the homomorphic extension of a m a p v E AX, and w E P ( X ) . T h e n G ( v ) = v(w).

w. For x E X we have 2 ( v ) = ~ ( x ) ( v=) a,(v) = v(z) = C ( x ) . If w1,...,wT(x) E P ( X ) satisfy the property then Proof by algebraic induction on

-

Fxwl*"wT(x)(v)

=

= f x ( G 1 > *'.,GT(x))(v)=

fx(61(v),"',tZT(x)(v))

= fx(v(wl)>

.*.?V(wT(X)))

=

.

= ij(F~Wl...W,(~))

0

5.37. Proposition.

Let P ( X ) be a Peano algebra, A a r-algebra and w1,w2 E P ( X ) . T h e n

Proof. Note that every cp E Hom(P(X),A) is of the form

v E AX, namely v = 'pix. Therefore Lemma 5.36 implies

'p

= 6 where

69

Universal algebra

@ GI(.)

(V.)

= tq.)

.

0

At this point note that when we study a specific algebraic structure we are in fact interested i n its polynomials rather t h a n in t h e choice of t h e basic operations. Thus e.g., no matter whether an Abelian group is viewed as an algebra of type (2,1,0) or of type (2), i ts polynomials are t h e operations of

the form p ( t l , ...,zn) = t y ...22,where al,...,a, E

Z.

In this example th e basic operation of type (2) is one of t h e basic operations of type (2,1,0), b u t this feature is irrelevant, as it is missing in the next example. It is well known th a t every Boolean algebra becomes an idempotent (i.e., x 2 = t for all

(5.15')

5

t) ring

+ y = (2 A y) V ( 5 A y) ,

w i th unit if one defines

zy = 2 A y

and conversely, every idempotent ring w i th unit is a Boolean algebra under the operations

moreover, this correspondence is one-to-one.

Let us set this point i n t o a

general framework.

5.38. Definition. T w o algebras

A and A' o n th e same underlying set A (but not necessarily

of t he same type) are termed equivalent if they have the same polynomials.

5.39. Definitio n . A class K: of algebras of type T and a class K:' of algebras of type r' are said to be equivalent if there exist tw o maps @ : K + K' and !D : K:' --t K such that:

(i) every A E

K: is equivalent

to

@ ( A )and every A' E K' is equivalent to

@(A'),and (ii) * @ ( A )= A for every A E K: and @@(A') = A' for every A' E K:'.

70

Lattices, universal algebra and categories Now we can present a construction which is a sufficient condition for the

equivalence of two equational classes o f finite types. 5.40. Remark. Let

T

: {1,...,m } +

IV

be a finite type and

X

a non-empty set. Every

w E P ( X ) has a formative construction which uses (some of) t h e symbols Fl,...,F,; we indicate this by writing w = w(Fl,..., Fm).Further let ( A ,{ f i } + ~ ) be a T-algebra. Inasmuch as'is a homomorphism, it follows easily by algebraic induction that the polynomial G has a formative construction which uses fi, ..., f, in t h e same way as the construction of

Fl,...,F, have been used in

w ;we indicate this fact (which can be given an obvious

technical definition) by writing G = G(f1,

...,f m ) .

5.41. Notation. Let

T

and

(T

be two types o f algebras and X a non-empty set. Let P o ( X )

and P T ( X )denote the corresponding Peano algebras; their elements are denoted by w and w' respectively, possibly provided with indices. Finally let

K' be equational classes of .r-algebras and : K: +K:'and Q : K : ' + K : .

and

K:

o-algebras, respectively. Let

The next proposition uses the above notation. 5.42. Proposition.

Suppose T : (1, ...,m } + IV and (T : (1, ...,n } + IV are finite t y pes and Ax K: = {wlh, w z h ) / h E H } , Ax K:' = { w i k ,w i k ) / k E I<} f o r finite H and I<. Suppose further there exist w1, ...,w, E P o ( x ) and w;, ...,w:, E P T ( X )such that:

(i) f o r a n y A = ( A , { f ; } i = ~ E)

K: the operations

are of arities a(l), ...,( ~ ( n respectively, ), and satisfy (5.17)

a' W,

(gl,..., g,) = fi

(i = 1,...,m ) ,

71

Universal algebra (5.18)

m’ Wlk

where” and

m

(Sl,...,gn) =EL. (g1, ...,gn)

(Vk E I.’)

denote the canonical morphisms of the algebras A and

@(A)= ( A , {gj}j=F), respectively, and (ii) for a n y A’ = ( A , {gj}j=F) E K’ the operations (5.19)

fi

= Gi(g1,

...,gn) (i = 1,...,m )

are of arities .r(l), ...,~ ( r n )respectively, , and satisfy

where- and k: denote the canonical morphisms of the algebra A’ and Q(A‘)= ( A , { f ; } i = ~ )respectively. ,

Then the classes K and K’ are equivalent via the transformations @ and Q. Proof. For every therefore @ :

K

A E Ic, @(A)is --f

a .r-algebra and

K’. It follows easily from (5.16)

@ ( A )E K’ by (5.18), that the polynomials of

+ ( A ) are polynomials of A, while (5.17) implies t h a t t h e polynomials of A are polynomials of @(A), therefore A and * ( A ) are equivalent. One proves similarly that Q : K’ + K and A’ and Q(d’)are equivalent for every A’ E K’. It follows readily that Q @ ( d=) A and @*(A‘)= A’. 0 5.43. Definition.

The situation described in Proposition 5.42 will be expressed simply in the following terms: every algebra of K is equivalent t o a n algebra of K’ via the transformations (5.16) and (5.19). Thus the example before Definition 5.35 can be reworded t o the effect that every Boolean algebra is equivalent to a Boolean ring with unit via the transformations (5.15’) and (5.15”). in the subsequent chapters.

We will actually use Proposition 5.42

Lattices, universal algebra and categories

72

We conclude this section with a universal-algebraic construction which is important in logic. 5.44. Definition.

be a family of .r-algebras (or simply sets) and E a filter of the Boolean algebra ( F ( H ) , s ) . Let further A = II Ah be t h e direct Let

(&,)hEH

hEH

product of t h e algebras Ah and -F- the relation on A defined by

5.45. Lemma. -F- is a congruence on

A.

Proof. Routine. 5.46. Definition.

If

is an ultrafilter on

0

( P ( H ) ,c)then t h e quotient algebra A / -F, -

denoted

also A / E , i s called t h e ultraproduct of the algebras Ah with respect t o E.

73

Categories and f u n c t o r s

56. Categories and functors In this section we list those category- heoretical concep s and results that will be used in the book. Proofs are omitted.

6.1. Definition. A category C is a class ObC endowed with the following structure: (i) For any A, B E ObC there is a set C(A, B ) such that (A,B)#(A‘,B’)

+ C(A,B)nC(A’,B‘)

=8;

(ii) For any A, B , C E ObC and for any f E C(A, B ) , g E C(B, C ) there is a unique element g o f of the set C(A,C) such that the following conditions are verified:

f E C(A,B), g E C(B, G ) , h E C(C, 0) f) = ( h o g ) o f ;

(a) If A, B, C,D E ObC and

then h o (g o

(b) For any A E ObC there is an element 1~ in C(A,A) such that 1~ 0 f = f and g o 1 A = g, for any B,C E ObC, f E C ( A , B ) and g E C(A,C). The elements of ObC are called objects of C and an element f of C ( A , B ) is called a m o r p h i s m from A t o B. If f E C(A,B) then we shall write f : A + B or A & B ; A is the d o m a i n o f f and B t h e codomain of f ; g o f is called the composite o f f and g. Thus composition is a partially defined associative operation. The element 1~ is called t h e identity of A . We shall write A E C instead of A E ObC and sometimes g f instead of g o f . It is easy t o verify the uniqueness of the identity. A category is usually described by mentioning first i t s objects, then i t s morphisms. Thus e.g.:

6.2. Definition. In what follows Set will be the category of sets and functions and Top the

74

Lattices, universal algebra and categories

category of topological spaces and continuous functions. If K: is a class of similar algebras then K will be the category whose objects are t h e algebras in K and whose morphisms are all t h e homomorphisms f

: A -+ B , where

A,B E K . K will be called an algebraic category. Thus Po is the category of posets and isotone mappings, DO1 is th e category of bounded distributive lattices and bounded-lattice homomorphisms, Mg is the category of De Morgan algebras and De Morgan homomorphisms, B is the category of Boolean algebras and Boolean homomorphisms. An equational category is an algebraic category K where

K is a variety. K is trivial if cardA

= 1 for any

A E K. 6.3. Definition. The d u d category Co o f a category C is defined by ObC' = ObC, C o ( X ,Y )= C ( Y , X ) for any X , Y E Co and for any morphisms f : X 4 Y and g : Y 4 2 , t h e composite g o f in C" is the composite f o g in C. 6.4. Definition.

A subcategory of C is a category C' such that (i)

ObC' E ObC;

(ii) C'(A,B)5 C(A,B)for any A,B E C'; (iii) For any A E C' the identity of A in C' coincides with the identity of

A in C . (iv) The composition of morphisms in C' is the same as in C.

C' is a fuZ1 subcategory of C if to conditions (i)-(iv) we add (v)

C'(A,B ) = C(A,B ) for any A, B E C'.

6.5. Definition. A morphism f of a category C is a monomorphism (an epimorphism) if f o g = f o h implies g = h (if g o f = h o f implies g = h ) for any morphisms g , h in C.

Categories and functors A morphism f : A

75

B of C

isomorphism if there is a morphism g : B + A in C such that g o f = 1 A and f o g = 1 B ; the morphism g is unique and is denoted by f - l . Then f-’ i s also an isomorphism. The 4

is an

composite of two isomorphisms is an isomorphism. Every isomorphism is a monomorphism and an epimorphism, but the converse is not true. Two objects A , B are isomorphic if there is an isomorphism f : A -+ B ;

in this case, we will write A

E

B.

B E C if there exist two morphisms f : A --t B and g : B 3 A such that g o f = 1 A . A variant of this definition i s to say that A is a retract of f : A 4 B if there i s g : B -+ A such that g o f = 1 A . An object A E C is a retract of an object

Note t h a t if g o

f = 1~

then

f is a

monomorphism and g is an epimorp-

hism.

6.6. Proposition (Balbes and Dwinger [1974]). Let K be a n equational category and f E K(A, B ) . T h e n f is a m o n o -

morphism ;$ f is injective.

6.7. Definition. Let A be an object of a category C. A subobject of A is a monomorphism

f

B + A ; sometimes we will abusively say t h a t B is a subobject of A . f2 : B” -+ A are isomorphic if there is an isomorphism g : B’ -+ B” such t h a t f2 o g = f l , :

Two subobjects f1 : B‘ + A,

6.8. Definition. An eztension of an object A is a monomorphism f : A -+ B. An extension f’ : A + B‘ is included in (isomorphic to) an extension f ” : A --t B” if there is a monomorphism (an isomorphism) g : B’ -+ B” such that g o f‘ = f”. An extension f : A --t B is called proper if f is not an isomorphism and essential if for any morphism g : B -+ B’, if g o f is a monomorphism then g is a monomorphism.

76

Lattices, universal algebra and categories

6.9. Definition. Let {Ai}iE~ be a family of objects in a category C. A direct product of

{Ai}iEl is a family {ri : A + A;}i,g of morphisms in C with the property t h a t for every family {f; : B + A i } i E l of morphisms in C there is a unique morphism f : B + A such that ri o f = fi for any i E I . The morphisms r; are called projections. 6.10. Remark. In any equational category K the direct product of a family { A i } i € ~is the Cartesian product

II Ai endowed with the canonical structure of an alge-

iEI

bra in

I<.

6.11. Definition. Let {Ai}iEI be a family of objects in a category C. A direct sum (coproduct) of { A i } i E if ~ a family {ji : Ai + A}i,I

of morphisms in C with the property

Ai + B};,I of morphisms in C there is a unique morphism f : A --t B such that f o j ; = fi for any i E I . The morphisms ji are known as injections (although they need not be injective functions). that for every family

{fi:

6.12. Remarks. (a) The direct product of a family {Ai}iEI is unique up t o an isomorphism:

if {xi : A + Ai}iEl, {r: : A’ + Ai}iEl are two direct products of { A i } i E then ~ there is an isomorphism f : A + A’ such t h a t r: o f = ri for any i E I . Sometimes we shall say that the direct product of {Ai}iE~ is A and write A = II A;. &I

(b) The direct sum of a family ( A i } i € ~is unique up t o an isomorphism. If

{ji : A; + A}iEl is the direct sum of { A i } i € ~we also say that the Ai.

direct sum of {Ai};€l is A and we write A =

i€Z

Proving the existence of direct sums is in general a difficult problem. For

77

Categories and functors algebraic categories we have the following result:

6.13. Theorem (Pierce [1968], Corollary 4.2.8). If K is a n algebraic category closed t o direct products and subalgebras and {A;}iE1 is a f a m i l y of objects in K such that f o r each i E I , Ai is a subalgebra of B;, f o r s o m e B; E K and there exist morphisms h;,j : A; t Bj f o r each pair i , j , t h e n the coproduct of

{Ai}iEl

ezists.

6.14. Definition.

If C, D are two categories, then a covariant f u n c t o r or simply a f u n c t o r F : C -+ D is a map which assigns t o any A E ObC an object F ( A ) of D and t o any morphism f : A t B in C a morphism F ( f ) : F ( A ) 4 F ( B ) in D, such t h a t

(b) F ( g o

f) = F ( g ) o F ( f ) , for

any morphisms f : A

+ B,

g : B +C

in C.

A contravariant f u n c t o r F : C t D is a map which assigns t o any A E UbC an object F ( A ) of D and t o any morphism f : A -+ B in C a morphism F(f) : F ( B ) + F ( A ) in D such t h a t (a')

F ( ~ A=) ~ F ( A )for , any A E ObC; ,g : B

t

C

For every category C we will denote by idc the identity f u n c t o r idc : C

--t

C

(b') F ( g o f) = F ( f ) o F ( g ) for any morphisms f : A in C. defined by idc(A) = A and idc(f) =

f E C ( A , B ) . If F

D

f

and G

+B

for any A , B E ObC and any

D

E are two covariant (contravariant) functors, their composite G o F : C t E is defined by (G o F ) ( A )= G(F(A)) and (G o F ) ( f )= G(F(f)) for any A , B E ObC and any f E C(A,B). : C

-+

:

t

It is easy t o see that: the composite of two covariant (contravariant) functors is a covariant functor, composition is associative, F o idc = F for

Lattices, universal algebra and categories

78

F : C

any functor

--t

D

and idc o G = G for any functor G :

D + C.

6.15. Defi nitio n . A covariant functor F : C + C' is fulZ (faithful, fully faithful) if, for any

A , B E C , the map

F : C ( A ,B ) + C ' ( F ( A ) , F ( B ) ) is injective (surjective, bijective). A similar definition is given for t h e contra-

F preserves (reflects) a property P if F(f)(if f ) satisfies P whenever f (whenever F ( f ) ) has property P.

variant functors. A covariant functor

6.16. Definition. A covariant functor F : C + D is an isomorphism if there is a covariant functor G : D + C such t h a t G o F = idc and F o G = idD. This concept being t o o restrictive, one usually prefers t h e concept of

equ ivaIe nce.

6.17. Defi nitio n . A covariant functor F : C + D is an equivalence of categories if it is fully faithful and for any B E D there is A E C for which F ( A ) is isomorphic to B. We say t h a t the categories C and D are equivalent. 6.18. Remarks. (a) Any equivalence

F : C + D preserves and reflects monomorphisms,

epimorphisms, direct products and direct sums.

(b) A contravariant functor F : C + D defines in a natural way a covariant functor

F* : C + Do. F is a coequivalence of categories if F* is an

equivalence o f categories. In this case, t h e category D is equivalent t o t h e dual category o f

C.

6.19. Definition. Let F

:

C + D, G

:

D

--f

C be t w o covariant functors. A

na-

79

Categories and functors

tural transformation or a functorial m o r p h i s m X : F 4 G is a family {XA : F ( A ) t G ( A )I A E C} of morphisms of D such that for any morphism f : A -+ B in C the diagram in Fig. 6.1 is commutative. We say that X is natural in A.

6.20. Notation. Let C be an arbitrary category and X E the covariant functor defined by:

C. We will denote by hX : C -+ Set hX(Y)= C(X,Y), Y E C and for

Y + 2 , h x ( f ) is the function given by h x ( f ) ( g ) = f o g, for E C(X,Y). The contravariant functor h , : C + Set is defined by: h x ( Y )= C(Y,X), Y E C and for f : Y 4 2 , h x ( f ) is the function given

f

:

any g

by hx (f>(s) =9

0

f,for any 9 E C(2,XI.

6.2 1. Def inition. Let F : C -+ D , G : D t C be two covariant functors. F is a Zeft adjoint of G and G is a right adjoint of F if for any X E C ,X’ E D there is a bijective map:

cp(X,X‘): C(X,G(X‘)) + D(F(X),X’) such t h a t for every morphisms

f : X

--f

the following diagrams are commutative:

Y

in

C and f’ : X’+ Y’in D

Lattices, universal algebra and categories

80

6.22.Remarks. X

(a) Let us consider the morphisms qx :

X‘

--+

G ( F ( X ) ) ,EX’ : F ( G ( X ’ ) ) --t

defined by qx = cp(X, F ( X ) ) - * ( l F ( X ) )E,X ! = (p(G(X‘),X’)(lc(x,)).

We can see that qx is natural in X and E X : is natural in X’, therefore we have the natural transformation q : idc -+ G o F and E : F O G4 idD. q and e are called adjointness morphisms associated with F and G. (b) Let C‘ be a subcategory of C. C‘ is reflective if t h e inclusion functor C’ + C has a left adjoint, which is called the refEector.

6.23. Proposition.

If q, e are the adjointness morphisms G

:

of the adjoint functors F : C + D ,

D -+C and X E C , X‘ E D then

81

Categories and functors

6.24. Proposition. Let F : C + D be a covariant functor and G a right adjoint of F . Then F preserves direct sums and epimorphisms, while G preserves direct products and monomorphisms. 6.25. Proposition. Let G : D + C be a right adjoint of F : C (a)

-4

D. Then:

The functor F as faithful iflprlx is a monomorphism for each X E C.

(b) G is fully faithful i f

EXI

is an zsomo~phismfor each X’ E D.

6.26. Proposition. For any covariant functor F

:

C + D the following assertions are

equivalent (i)

F is an equivalence of categories;

(ii) F is fully faithful and it has a fully faithful left adjoint; (iii) F is fully faithful and it has a fully faithful right adjoint. The proofs of Propositions 6.24-6.27 can be found in MacLane I19711 or Popescu and A. Radu [1971]. 6.27. Definition. An object A of a category C is called injective if for any monomorphism f : B + C and for any morphism g : B + A there is a morphisrn h : C + A such that h o f = g . 6.28. Definition. An object A of C is called projective if for any epimorphisrn f : C -+ B and for any morphism g : A --t B,there is a morphism h : A --f C such that f o h = g .

32

Lattices, universal algebra and categories

6.29. Remarks. (a)

A retract of an injective (projective) object is also injective (projective).

(b) A n injective object A is a retract of any of its extensions A (take B = A , C = A, f : A + A and g = 1~in Definition 6.28). (c) Any direct product (sum) of a family of injective (projective) objects is also injective (projective).

6.30. Definition. An

injective hull of A E C is an injective essential extension of A .

6.31. Remark. In every equational category any t w o injective hulls of an object are isomorphic.