REPRESENTATION THEOREMS IN UNIVERSAL ALGEBRA AND ALGEBRAIC LOGIC

by

Martin Izak Pienaar

THESIS

presented in partial fulfillment of the requirements for the degree

MAGISTER SCIENTIAE

in

MATHEMATICS

in the

FACULTY OF SCIENCE

of the

RAND AFRIKAANS UNIVERSITY

Study leader: Prof. V. Goranko

NOVEMBER 1997 Baie dankie aan:

Prof. V. Goranko - vir leiding in hierdie skripsie.

RAU en SNO - finansiele bystand gedurende die studie.

my ouers - onbaatsugtige liefde tydens my studies.

Samantha Cambridge - vir die flink en netjiese tikwerk.

ons Hemelse Vader - vir krag en onderskraging in elke dag.

Vir my ouers en vriende - julie was die .inspirasie viz- my studies CONTENTS

1 INTRODUCTION TO LATTICES AND ORDER 1

1.1 Ordered Sets 1 1.2 Maps between ordered sets 2 1.3 The duality principle; down - sets and up - sets 3 1.4 Maximal and minimal elements; top and bottom 4 1.5 Lattices 4

2 IMPLICATIVE LATTICES, HEYTING ALGEBRAS AND BOOLEAN ALGEBRAS 8

2.1 Implicative lattices 8 2.2 Heyting algebras 9 2.3 Boolean algebras 10

3 REPRESENTATION THEORY INVOLVING BOOLEAN ALGEBRAS AND DISTRIBU- TIVE LATTICES: THE FINITE CASE 12

3:1 The representation of finite Boolean algebras 12 3.2 The representation of finite distributive lattices 15 3.3 Duality between finite distributive lattices and finite ordered sets 17

4 REPRESENTATION THEORY INVOLVING BOOLEAN. ALGEBRAS AND DISTRIBU- TIVE LATTICES: THE GENERAL CASE 20

4.1 Ideals and filters 20 4.2 Representation by lattices of sets 22 4.3 Duality 27

5 REPRESENTATION THEORY: A MORE GENERAL AND APPLICABLE VIEW 30

5.1 Analogue statements for the representation theorems of chapters 3 and 4 30 5.2 Introduction to relation algebras, finite - dimensional cylindric algebras, and their interconnections 32 5.3 Representation of relation algebras 35 5.4 Representation of finite - dimensional cylindric algebras 37

6 BOOLEAN ALGEBRAS WITH OPERATORS'- , 42

6.1 Definitions and notation 43 6.2 The so - called extension theorem for Boolean algebras with operators 44 6.3 A representation theorem for Boolean algebras with operators 45 6.4 Application of the results on Boolean algebras with operators to closure algebras and cylindric algebras 47

7 SOME APPLICATIONS OF REPRESENTATION THEOREMS IN UNIVERSAL AL- GEBRA AS WELL AS IN ALGEBRAIC LOGIC 51

7.1 Applications in universal algebra 51 7.2 The subdirect product representation theorem - an extension of Birkhoff's repre- sentation theorem for finite distributive lattices 53 7.3 Applications in algebraic logic 55

REFERENCES 59 OPSOMMING

In hierdie skripsie word die invloed van sg. "representation theorems" in universele algebra en algebraiese logika, op sekere abstrakte wiskundige strukture beskou. Stellings wat 'n sekere abstrakte wiskundige struktuur voorstel as 'n meer algemene wiskundige model, word ondersoek. Die skripsie is as volg georganiseer:

Hoofstuk 0 dien as 'n basiese inleiding tot "representation theorems". Dit verduidelik die idee agter "representation theorems" en verskaf 'n basiese "representation theorem".

.Hoofstuk 1 bestaan hoofsaaklik uit definisies wat van belang is vir die res van die skripsie.

In hoofstuk 2 word die wiskundige strukture, naamlik "implicative lattices", "heyting alge: bras", en "Boolean algebras" ondersoek.

Hoofstuk 3 verduidelik die rol van "representation theorems" in eindige "distributive lattices" en eindige "Boolean algebras".

In hoofstuk 4 word ekwivalente resultate as die van hoofstuk 3, vir arbitrere "Boolean alge- bras" en "distributive lattices" bespreek.

Hoofstuk 5 bespreek die idee van "representation theorems" in die algemeen. Twee verdere strukture, naamlik "relation algebras" en "finite - dimensional cylindric algebras" word be- spreek.

Hoofstuk 6 is 'n "uitbreiding" van vorige resultate. Die rol van "representation theorems" in arbitrere "Boolean algebras" word bespreek.

In hoofstuk 7 word sommige toepassings van "representation theorems" in universele algebra en algebraiese logika beskou.

-000- SUMMARY

The objective of this thesis is to discuss the essence and importance of representation the- orems in the area of universal algebra and algebraic lOgic, generally and by means of some fundamental examples.The thesis is organized as follows:

Chapter 0 is introductory. It gives an initial illustration of the topic by means of one of the historically first and most important examples of representation theorems in mathematics, viz. Cayley's representation theorem for groups.

Chapters 1 and 2 contain preliminary material on order and lattices as well as specific types of lattices, Heyting and Boolean algebras, which are important in algebraic logic. They provide the neccessary technical background for the thesis.

Chapters 3 to 7 are the core of the thesis.

Chapter 3 presents in detail the representation theorems for finite distributive lattices via duality with ordered sets and for Boolean algebras as fields of subsets of finite sets.

Chapter 4 presents the classical representation theorems(in the general case) for distributive lattices and Boolean algebras, due to M.Stone, G.Birkhoff and H.Priestley, and discuss the duality between Boolean algebras and Boolean topological spaces.

Chapter 5 mentions analogous results for implicative lattices and Heyting algebras due to Stone and Rasiowa-Sikorski. Further, it briefly introduces relation algebras and some basic results about them.

Chapter 6 is based on the classical result of Tarski and Jonsson providing a general represen- tation theorem for Boolean algebras with normal and additive operators, with applications to representation theorems for relation algebras and cylindric algebras.

Chapter 7 mentions some representation results in universal algebra., most importantly Birkhoff's subdirect reducibility theorem. It also briefly discusses an application of rep- resentation theorems in algebraic logic, illustrating the case of intuitionistic logic.

-000- Chapter 0

Introduction to Representation Theorems

Quite often, all the different fields in Mathematics seem to be independent with no inter- connection. In a specific field, for instance algebra, certain structures sometimes seem to be very abstract and it is difficult to express the global effect of the involved structure in that field. This "problem" is partly overcome by so - called representation theorems. By a representation theorem, the "representation" of an abstract mathematical structure as Xsometimes well - known) mathematical model is studied. This ensures that the involved structure as well as its applications become clearer. Because representation theorems are mostly based on isomorphisms between the involved structures, one can also conclude that properties which hold in the one structure must also hold (up to an isomorphism) in the other, less - involved, structure.

To explain the ideas mentioned in the above paragraph, it is good to investigate one of the historical first examples of a representation theorem. We are namely going to state and prove Cayley's well - known theorem which ensures that an arbitrary group is isomorphic to a group of permutations. This would mean that] properties which hold in a certain group, will also be true in the involved group of permutations (due to the isomorphism between the two structures).

Definitions:

A permutation of a set A is a function from A to A that is both one - to - one and onto.

A permutation group of a set A is a set of permutations of A that forms a group under function composition.

Informally, Cayley's theorem states that every group is isomorphic to some group consisting of permutations under permutation multiplication. This is a nice and intriguing result, and is a classic in group theory. At first glance, the theorem might seem to he a tool to answer all questions about groups. What it really shows is the generality of the groups of permutations. Examining subgroups of all permutation groups for a set of all sizes would be a tremendous task. Cayley's theorem does show that if a counterexample exists to some conjecture you have made about groups, then some group of permutations will provide the counterexample.

We next state and prove the actual theorem:

Theorem 0.0.1 Cayley's Theorem Every group is isomorphic to a group of permutations.

Proof:

Let G be a given group.

Step 1: Our first task is to find a set C' of permutations that is a candidate to form a group isomorphic to G. Think of G as just a set, and let SG be the collection of all permutations of G. Then SG is a group under permutation multiplication (this fact follows easily from basic definitions). Note that in the finite case if C has n elements, SG has n! elements. Thus, in general, SG is clearly too big to be isomorphic to G. We define a certain subset of SG. For a E C, let A, be the mapping of G into G given by Aa (x) = ax for x E G. (We can think of )1„ as left multiplication by a). If A a (s) = A a (y), then ax = ay so that x = y (by an easy exercise). Thus A, is a one - to - one function. Also, if y E G, then A a (a - ly) = a(a'y) = y, so )' a maps G onto G. Since Aa : G is both one - to - one and onto G, A, is a permutation of G; that is A, E SG. Let now G' = a E G}.

Step 2: We claim that G' is. a subgroup of SG. We must show that G' is closed under Permutation multiplication, contains the identity permutation, and contains an inverse for each of its elements. First, we claim that )t a \b = Aab• To show that these functions are the same, we must show that they have the same action on each x E G. Now (A,Ab)(x) = A a (Ab(x)) = A a(bx) = a(bx) = (ab)x = Aab(x). Thus Adtb = kb, so G' is closed under multiplication. Clearly for all x E G, ) e (x) = ex = x, where e is the identity element of C, so A, is the identity permutation i of SG and is in Since A„A b = A ab, we have A„A„-i = A aa -i = Ae , and also A„--1A„ = )' e . A: 1 , Hence (Aa)- so that (A,) -1 E Thus G' is a subgroup of SG. Step 3: It remains for us to prove now that G is isomorphic to this group G' that we have described. Define : G C' by 0(a) = Aa for a E G. If 0(a) 0(b), then .\a and )'b must be the same permutation of G. In particular, A a (e) = )t b (e), so ae = be and a = b. Thus 0 is one - to - one. It is immediate that 0 is onto G' by the definition of G'. Finally, 0(ab)= Aabi while q(a)q(b) = Aa Ab. But we saw in step 2 that )tab and )'a )b are the same permutation of G. Thus 0(ab) 85(a)0(b), so that 0 is an isomorphism from G onto G'. Therefore the theorem holds.

Note that, for the above proof, we could have considered equally well the permutations p a of G defined by pa(x) = xa for x E G. (We can think of p a as meaning right multiplication by a). These permutations form a subgroup G" of SG, again isomorphic to G, but under the map it : G G" defined by it(a) = pa-i. The group G' in the proof of Cayley's theorem is the left regular representation of G, and the group G" in the preceeding comment is the right regular representation of G. Chapter 1

Introduction to Lattices and Order

1.1 Ordered Sets

Definitions: Let P be a set. An order (or partial order) on P is a binary relation < on P such that, for all x, y, z, E P,

x 5_ x,

x < y and y < x imply x = y,

x < y and y < z imply x < z.

These conditions are referred to, respectively, as reflexivity, anti-symmetry and transitivity. A set P equipped with an order relation < is said to be an ordered set (or partially ordered set - also known as a "poset"). Where it is necessary to specify the order relation, we write (P; An order relation < on P gives rise to a relation < of strict inequality: x < y in P if and only if x < y and x y. It is possible to re-state conditions (i) - (iii) above in terms of <, and so to regard < rather than < as the basic relation.

Other notation associated with < is predictable. We use x < y and y > x interchangeably, and write x y to mean x < y is false, and so on. Less familiar is the symbol II used to denote non- comparability: we write x y if x y and y X x.

1.1.1 Chains and antichains

Let P be an ordered set. Then P is a chain if, for all x, y E P, either x < y or y < x. Alternative names for a chain are linearly ordered set and totally ordered set. The ordered set P is an antichain if x < y in P only if x = y. Clearly, with the induced order, any subset of

a chain (an antichain) is a chain (an antichain). Let P be the n-element set {0,1, ,n —1}. We write n to denote the chain obtained by giving P the order in which 0 < 1 < < n 1 and n for P regarded as an antichain.

1.1.2 The covering relation.

Let P be an ordered set and x, y E P. We say x is covered by y (or y covers x), and write x y or y F- x if x < y and x < z < y implies z = x.The latter condition is demanding that there be no element z of P with x < z < y.

Observe that, if P is finite, x < y if and only if there exists a finite sequence of covering relations x = xo H x i H ... = y. Thus, in the finite case, the order relation determines, and is determined by, the covering relation.

1.2 Maps between ordered sets

Definitions: Let P and Q be ordered sets. A map cp:P---)Qis said to be

order-preserving (or, alternatively, monotone) if x < y in P implies cp(s) < cp(y) in Q;

an order-embedding if x < y in P if and only if cp(x) < v(y) in Q;

an order-isomorphism if it is an order-embedding mapping P onto Q.

When cp : P Q is an order-embedding we write cp : Q. When there exists an order- isomorphism from P to Q, we say that P and Q are order-isomorphic and write P Q.

Remarks:

Let cp : P Q and : Q R be order- preserving maps. Then the composite map o v, given by (0 o c,9)(x) 1/)(yo(x)) for x E P, is order-preserving. More generally the composite of a finite number of order-preserving maps is order-preserving, if it is defined.

Let cp : P Q and let y(P) (defined to be {cp(x) x E P}) be the image of cp. Then V(P) P. Ordered sets P and Q are order-isomorphic if and only if there exist order-preserving . maps y : P Q and ?,/, : Q P such that cp. o 7b= idQ and o cp = idp (where ids : S S denotes the identity map on S given by ids (x) = x for all x E S).

9 1.3 The Duality Principle; down-sets and up-sets

1.3.1 The dual of an ordered set

Given any ordered set P we can form a new ordered set P8 (the dual of P) by defining x < y to hold in P6 if and only if y < x holds in P. For instance, if there exists a unique element in P which covers just 3 other elements, then there exists a unique element in Pb which is covered by just 3 other elements. This situation is illustrated in figure 1.1.

In general, given any statement (I) about ordered sets, we obtain the dual statement (D6 by replacing each occurrence of < by > and vice versa.

1.3.2 The Duality Principle

Given a statement (I) about ordered sets which is true in all ordered sets, then the dual statement 4)6 is true in all ordered sets.

1.3.3 Definitions and Remarks

Let P be an ordered set and Q C P.

Q is a down-set (alternative terms include decreasing set or order ideal) if; whenever xE Q,yEP and y < x, we have y E Q

Dually, Q is an up-set (alternative terms are increasing set or order filter) if, whenever xE Q,yEP and y > x, we have y E Q.

3 Besides being related by duality, clown-sets and up-sets are related by complementation: Q is a down-set if and only if P \ Q is an up-set.

Given an arbitrary subset Q of P and x E P, we define

Q = {yE P (3x E Q)Y < *x}; Q = {y E P (3x E Q)y > x}; x = {yEP I y5x} and x = {y€P I y>x}.

It is obvious that I Q is the smallest down-set containing Q and that Q is a down-set if and only if Q =j, Q, and dually for I Q. Clearly 1 {x} x, and dually for x. The family of all down-sets of P is denoted by•V(P). It is itself an ordered set, under the inclusion order, and plays a crucial role in the proofs of representation theorems involving lattices and order.

1.4 Maximal and minimal elements; top and bottom

1.4.1 Maximal and minimal elements

Let P be an ordered set and let Q C P. Then

a E Q is a maximal element of Q if a

a E Q is the greatest (or maximum) element of Q if a > x for every x E Q, and in that case we write a = max Q.

A minimal element of Q, the least (or minimum) element of Q and min Q are defined dually, that is by reversing the order. Note that the greatest (or least) element need not exist, but if it does it is unique.

1.4.2 Top and bottom

Let P be an ordered set. The greatest element of P, if it exists, is called the top (element) of P and written T. Similarly, the least element of P, if it exists, is called the bottom (element) and denoted by 1.

1.5 Lattices

Definitions:

4 (i) Let P be an ordered set and let S C P. An element x E P is an upper bound of S if s < x for all s E S. A lower hound is defined dually. The set of all upper bounds of S is denoted by Su and the set of all lower bounds by S'. Thus:

Su := E P I (Vs E S)s x} and S I := E P I (Vs E S)s x}.

(ii) Let (X, <) be a poset and x, y E X. The least upper bound of {x, y}, if it exists, is called the join of x and y, denoted by x V y; the greatest lower bound of {x, y}, if it exists, is called the meet of x and y, denoted by x A y.

(iii) First notion of a lattice: A lattices is a non-empty poset (X, <) in which every two elements have join and meet. EXamples of lattices 1 are, for instance:

Every linearly ordered set. (P(X),C) for any set X. (Ar+, I), where I is the divisibility relation in N.+. The family of subspaces of a linear space, ordered by C. The family of subgroups of a group, ordered by C.

(iv) Second notion of a lattice: A lattice 2 is a non-empty set X endowed with two binary operations, _A and V, which satisfy the following identities for every x, y, z E X:

xAx=x; xVx=x sAy=yAx; xVy=yVx x A (y A z) = (x A y) A z; x V (y V z) = (x V y) V z x V (x A y) = x; x A (x V y) = x

The above two notions of a lattice are equivalent in the following sense:

Proposition 1.5.1 (i) If A, V) is a lattice2 , then the relation < defined by x < y if and only if x A y = x is a partial order in X and (X, <) is a lattice' with a meet A and a join V.

(ii) If (X, <) is a lattices , then the corresponding lattice2 (X, A, V) satisfies the property x < y if and only ifsAy=x if and only if x V y = y for every x,y E X, and thus determines a lattice' identical with (X, <).

Proof:

(i) Let (X, A, V) be a lattice2 . The relation < in X, defined by x < y if and only if x Ay = x, is a partial ordering:

5 For every x E X, x A x = x hence x < x, so that < is reflexive. Suppose x < y and y < z, for x, y, z E X. Thus x Ay = x and y A z = y. Hence xAz= (x A y) Az = x A (y A z) = x A y = x, so that x < z which implies that < is transitive. Suppose x < y and y < x, for x, y E X. Thus x A y = x and y A x = y, so that x = y which implies that < is antisymmetric.

Furthermore, for every x,y E X,

(xAy)Ax=xA(yAx)=xA(sAy)=(xAx)Ay=xAy.

Likewise (x A y) A y = x A (y A y) = x A y, so that x A y is a lower bound for x and y. x A y is the greatest lower bound of x and y: Letz

zA(sAy).(zAx)Ay=zAy=z

which implies that z < x A y. Also, for every x,y E X,

(xVy)Vx=xV(yVx)=xV(xVy)=(xVx)Vy=xVy.

Likewise (x V y) Vy=xV (y V y) = x V y, so that x V y is an upper bound for x and y. x V y is the least upper bound of x and y: Let z > x and z > y, for z E X. This means that x < z and y< z, so that x A z = x and y A z= y. Therefore xVz =z and y V z = z. Then zV(xVy).(zVx)Vy=zVy=z which implies that z A (x V y) = x V y. Thus (x V y) Az = x V y, so that x V y < z. Thus: (X, <) is a non-empty poset in which every two elements have join and meet, so that (X, <) is a lattice'.

(ii) We just give a brief summary of the proof of this part: For every x, y E X, if x < y then x is the greatest lower bound of x and y and y is the least upper bound of x and y. Conversely, if x Ay = x then x < y and so forth.

Hereafter we identify the notions of lattice' and lattice 2 and call each of them a lattice.

Definitions:

6 (i) A lattice (X, <) is modular if it satisfies the following modular law: For every x, y, z E X,

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

Examples of modular lattices are:

Any linear ordering. (P(X), C ) for any set X. The lattice of subspaces of any linear space.

(ii) A lattice (X, A, V) is distributive if it satisfies either one of the following distributive laws: For every x, y, z E X,

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

Examples of distributive lattices are:

Any linear ordering. (P(X), ) for any set X.

(iii) A non-empty subset Y of a lattice X is a sublattice of X if it is closed under meets and joins, i.e. for every al, a2 E Y, a l V a2 E Y and al A a2 E Y.

1.5.1 Maps between lattices

Let (X1 , A1, VI) and (X2, A2, V2) be two lattices. A mapping h : Xl —44 2 is called:

A homomorphism (of lattices) if

h(x = h(s) A2 h(y) and h(x V 1 y) = h(x) V2 h(y) for every xl, x2 E

An isomorphic embedding if it is an injective homomorphism.

An isomorphism if it is a bijective (i.e. both injective and surjective (onto)) homomor- phism. Chapter 2

Implicative Lattices, Heyting Algebras and Boolean Algebras

In this chapter we begin the study of Boolean algebras qua lattices. The representation theory of chapters 2 and 3 carries this further.

2.1 Implicative Lattices

Definitions:

(i) Let (X, A, V) be a lattice and a, b, E X. An element c E X is called a pseudo-complement of a relative to b if c is the greatest element in X with the property c A a < b. We denote the pseudo - complement of a relative to b (if it exists) by a b. Note that:

For every x E X,x < a --+ b if and only if x A a < b. a a exists if and only if the lattice has a top element T and then a a = T.

(ii) A lattice (X, A, V) is an implicative lattice if a b exists for every a, b E X. The latter operator is called an implication.

Note that every implicative lattice has a top element T (but not necessarily a bottom element 1) and hence can be regarded as a system (X, A, V, T). Examples of implicative lattices are:

(P(X), C) for any set X.

(A representative example) The lattice of open dense subsets of a topological space. For any two elements, say A and B of this lattice,A B = Int (A U B) where Int(A U B) denotes the interior of the set AU B and A denotes the complementary set of A in the involved topological space.

8 In later chapters we will look more carefully into representation theorems involving distribu- tive and implicative lattices specifically. The following proposition will play a central part in these chapters.

Proposition 2.1.1 Every implicative lattice (X, A, V, T) is distributive.

Proof: Take any elements x, y, z E X. Denote u = (x A y) V (x A z). This means that x A y < u andxAz

2.2 Heyting Algebras

Definitions:

A Heyting algebra (also called a pseudo-Boolean algebra) is an implicative lattice which has a bottom element I.

If a is an element of a Heyting algebra then a 1 is called a pseudo-complement of a. We denote the pseudo-complement of a by

Note that:

For every x, x < ifandonlyif x Aa= l. Every Heyting algebra can be considered as an algebraic system (X, A, V, --+, T, satisfying certain conditions.

Examples of Heyting algebras are:

Every set lattice (.7-, n, u, --3,1).Note that in this lattice, ,F C P(X) for some set X;1 = cb and for any two subsets A and B of X, A B = (X — A) U B and = X — A. Every finite chain. (A representative example) The lattice of open sets of a topological space. For any two elements, say A and B, of this lattice, AAB=AnB and AVB=AU B.

9 2.3 Boolean algebras

Definitions:

Let X be a Heyting algebra and x E X. The pseudo-complement -ix is called a complement of x if x V -ix = T.

A Boolean algebra is a Heyting algebra in which every pseudo- complement is a corn- plernent, ie: x V -ix = T is an identity for every x an element of the involved Heyting algebra.

Many important identities can be proved to hold in every Boolean algebra. Some of these identities are stated in proposition 2.3.1.

Proposition 2.3.1 The following identities hold in every Boolean algebra:

x V —ix = T

= x

x y = -4 -IX

, (X A y) = -ix V -'y -'(x V y) = -ix A

x V y = A -iy)

x A y = V -iy)

(BAS) y) = x A -'y (x y) x = x

x y = -ix V y

Proof:

Since the proofs of the above propositions are all very similar, we just prove (BA2), (BA4) and (BA5):

We start with the proof of the following: a A b = 1 and a V b = T imply -'a = b.

IfaAb=lthen:

10 -'a V (a A b) (-'a V a) A (-ia V b) T A (-‘a V b) -iaVb; hence > b, and if a V b = T then

-'aA(aVb) (-ia A a) V (-ia A b) 1V(-iaAb)

hence < b. Thus: = b.

Now -ix A x = 1 and -ix V x = T; hence = x by applying the above result. This establishes (B2).

Finally (x V y) V (-i x A -'y) = x V [y V (-ix A -iy)] x V [(y V A (y V --T)] x V [(y V -ix) A T] xVyV = T V y = T

and (x V y) A ( -ix A -iy) = [x A (-ix A -iy)] V [y A (-a A -'y)] = [(x A -ix) A -'y] V [(y A -iy) A -ix] = A -iy) V (1 A -ix) = _L V _L = 1

Now, by applying the first result in this proof, -i(x V y) = A y which establishes (BA5). Similarly (by interchanging V and A, and 1 and T above), we can show that -i(x A y) = -a V which establishes (BA4).

The proofs of the other identities follow in a very similar way.

Boolean algebras are inextricably linked to logic. These links will become clear throughout the next chapters. Relevant representation theorems will also be studied in detail.

1 1 Chapter 3

Representation Theory involving Boolean Algebras and Distributive Lattices : the Finite Case.

In the previous two chapters we have introduced various classes of lattices. We now turn our attention to structure theorems involving some of these classes. In chapter 4 we give a con- crete representation, as a lattice of sets, of any (bounded) distributive lattice. This chapter deals less ambitiously , with the finite case and reveals a very satisfactory correspondence between finite distributive lattices and finite ordered sets. The proofs in chapter 3 and 4 follow mostly directly from B.A. Davey and H.A. Priestly: " Introduction to Lattices and Order".

3.1 The representation of finite Boolean algebras.

Definitions:

(i) Let G be a lattice with least element 0. Then a E G is called an atom if 0 -I a (i.e.: 0 is covered by a , which means the following: if x < y and x < z < y, then z = x). The set of atoms of r is denoted by .A(r). It may happen that a lattice has no atoms at all. The chain of non - negative real num- bers provides an example. Even a Boolean lattice (where we mean by a Boolean lattice a lattice ,C which is distributive, which has a least and greatest element 0 and 1 re- spectively, and of which every a E .0 has a unique complement a' E .C) may have no atoms. However, given any element a 0 in a finite lattice there exists an atom x such that 0 x < a. It is also very easy to prove that if B is a finite Boolean lattice then,, for each a E B, a = V{x E A(B) I a: < a} (where V{x i , x2 , ...} denotes the join of x i , x2 , ..., which means that we first determine x 1 V x2 (if it exists) whereafter we determine (x 1 V x2 ) V x3 (if it exists) and so on. Proceeding in this way we determine

12 the join of {x i , s2 , . The method used here is correct, since distributivity holds for the V - operation.).

(ii) Let be a lattice. An element x E £ is join - irreducible if:

x # 0 (in case r has a least element) x.aVbimpliesx=aorx=bforalla,bEr.

Condition (2) is equivalent to the more pictorial condition (2)': a < x and b < x imply aVb

In a chain, every non zero element is join - irreducible. Thus, if r is an n-element chain, then J(r) is an (n — 1)-element chain. In a finite lattice 4, an element is join - irreducible if and only if it has exactly one lower cover. This makes J(L) extremely easy to identify from a diagram of G. Figure 3.1 illustrates this. (Note that the join - irreducible elements are circled in every lattice).

13 The following lemma compares atoms and join - irreducible elements. It shows that in a Boolean algebra, 3(C) coincides with A(C).

Lemma 3.1.1 Let C be a lattice with a least element 0. Then

0 x in G implies x E 3(C);

if C is a Boolean lattice, x E 3(C) implies 0 H x.

Proof:

To prove (i), suppose by way of contradiction that 0 x and that x = a V b with a < x and b < x. Since 0 x, we have a = b = 0, whence x = 0 ... a contradiction.

Assume C is a Boolean lattice and that x E 3(C). Suppose 0 < y < x; we want y = 0.

We have x=xVy= (xV y) A ( —, y V y) = (x A -,y) V y. Since x is join - irreducible and

y < x, we must have that x = x A —1y, whence x < But then y = x Ay < Ay = 0, so y = 0 which proves (ii).

Theorem 3.1.2 The Representation Theorem for Finite Boolean Algebras

Let B be a finite Boolean algebra. Then the map 77 : a E A(B) x < a} is an isomorphism of B onto P(X), where X = A(B), with the ] of ?I given by 77 -1 (S) = V S for S E P(X).

Proof:

We first show that 77 maps B onto P(X). Let S = {a l , , ak } be a set of atoms of B and define a = V S. We claim S = ii(a). Certainly S C ii(a). Now let x be any atom such that x < a = a1 V ... V ak. For each i, we have 0 < x A ai < x. Because x is an atom, either x A ai = 0 for all i or there exists j such that x A ai = x.. In the former case, x = x A a = (x A ai) V ... V (x A ak ) = 0 ...a contradiction. Therefore x < ai for same j,. which forces x = ai , as a; and x are atoms. Hence n(a) C S, as we wished to show.

Before going on further, we mention the following result which can be easily proved: Let f : B C, where B and C are Boolean algebras.

(i) Assume f is a lattice homonorphism. Then the following are equivalent:

f (±) = and f (T) = T f = f (a)) for all a E B .

(ii) If f preserves then f preserves V if and only if f preserves A ... (*) .

14 Now, comparing the results in (*) above with our original definition for a lattice - homo- morphism, one can easily see that 77 is an isomorphism. Furthermore, let a, b E B. Then 77(a) C i1(b) implies that a = V n(a) < V ii(b) = b. It is trivial (by the transitivity of <) that n (a) C i(b) whenever a < b, so y is an order - isomorphism. Equivalently, one may conclude that y is an isomorphism of Boolean algebras.

As a corollary of the foregoing theorem, we state the following which can easily be proved to be equivalent: Let 13 be a finite lattice. Then

B is a Boolean lattice;

13 P(A(B));

13 is isomorphic to 2', for some n > 0.

3.2 The representation of finite distributive lattices

In the foregoing paragraphs we proved a very helpful representation theorem for finite Boolean algebras. We concluded up to some satisfactory equivalent results. Nevertheless, in the finite case one often gets the feeling that we are restricted to certain representations. There is little variety among finite Boolean algebras. As we shall see, finite distributive lattices as well as infinite Boolean algebras (in chapter 4) are much richer classes.

In the next section we show that a finite distributive lattice, G, is determined by the ordered set J(C) (or M(C)) alone. More significantly, the procedure for recapturing .0 from J(C) is relatively simple: it turns out that (r)). The resulting representation theory provides a very powerful tool for studying finite distributive lattices.

Lemma 3.2.1 Let r be a distributive lattice and x E .C, with x 0 in case G has a least element 0. Then the following are equilavent:

x is join - irreducible;

ifa,bEG and x

for any k E N, if ak E G and x < al V...Vak then x < at for some i(1 < i < k).

Proof:

(i) —> (ii): Assume x E J(L) and that a, b E G are such that x < a V b. We have

x = xA(aVb)...x

15 Because x is join - irreducible, x = x A a or x = x A b. Hence x < a or x < b, as required. (ii): This part follows easily from induction on k. The case k = 1 is trivial so that one gets the induction started at k = 2. (ii): This part also follows trivially. (ii) (i): Suppose the conditions in (ii) hold and that x = aV b. Then certainly x < a V b, soxaandx> b. Hencex=aorx=bsothat(i) holds.

Theorem 3.2.2 Birkhoff's Representation Theorem for Finite Distributive Lat- tices Let L be a finite distributive lattice. Then the map 77 : L 19(J(L)) defined by

77(a) = {x E J(G) I x < a} (= (C)n J. a) is an isomorphism of L onto 7 9(J(L)).

Proof:

Since < is transitive, it follows immediately that 77(a) E t9(3(G)). Furthermore, after com- paring ri with the results, (marked (*) ) which also appear in the proof of theorem 3.1.2 , it remains to show that 77 preserves order and that it is onto. Suppose a < b, for a, b E L. Bearing in mind that this is equivalent to say that j, a b and that (VQ E (p(P))b EQ .aEQ (where P is an ordered set), it follows trivially that 77(a) C 77(b). Let 77(a) C n(b). One can easily prove that

a = \{x E J(C) I x < a} for all a E and b E 3(L) I x b} for all b E L. It follows now that a = V 77(a) < V 77(b) = b. Finally, to prove that 71 is onto, let U E V(J(r) and write U = {a l , , ak). Define a to be al V ... V ak . We claim U = 77(a). To prove this, first let x E U, so x = ai for some i. Then x is join - irreducible and x < a, hence x E 7/(a). In the reverse direction, suppose x E 7/(a). Then x < a = al V ... V ak and lemma 3.2.1 implies x < ai for some i. Since U is a down - set and ai E U, we have x E U. Thus, Ti is order - isomorphism so that Ti is (equivalently) an isomorphism of L onto t9(„7(r)).

As an .inimediate corollary of the above theorem, one can conclude that, if G is a finite lattice, the following statements (of which the proofs follow trivially) are equivalent:

16 (1) .0 is distributive;

G 19 (.7(C)); .0 is isomorphic to a lattice of sets;

G is isomorphic to a sublattice of 2', for some n > 0.

Also note that, as proved in chapter 2, every implicative lattice is distributive. Therefore the theorems in the foregoing paragraphs also hold for finite implicative lattices. This is a powerful tool to bear in mind for later chapters.

Up to now we studied some very helpful representation theorems, involving Boolean algebras and finite distributive lattices, in this chapter. It became clear that the scope to work within finite distributive lattices is far more involved and applicable than to work with respect to finite Boolean algebras only. These ideas as well as some direct and indirect applications will be taken further in chapter 4 and later chapters. We finish off this chapter with some interesting insight in the duality between finite distributive lattices and finite ordered sets.

3.3 Duality between finite distributive lattices and fi- nite ordered sets

We first state the following result which follows easily since it is a natural companion to Birkhoff's representation theorem for finite distributive lattices: Suppose P is a finite ordered set. Then the map E : x x is an order - isomorphism from P onto J(t9(P)).

3.3.1 Finite distributive lattices and ordered sets in partnership

We denote by DF the class of all finite distributive lattices and by PF the class of all finite ordered sets. Theorem 3.2.2 as well as the remark in the above paragraph asserts that V(J(C)) and P J(i9(P)) for all e DF and P E PF. We call „I(C) the dual of and V(P) the dual of P.

By identifying a finite distributive lattice .0 with the isomorphic lattice O(I(C)) of down - sets of J(C), we may regard DF as consisting of the concrete lattices 19(P), for P E Pc", rather than as abstract objects satisfying certain identities.

Up to isomorphism, we have a one - to - one correspondence

(P(P) = C P = J(C) for .0 E DF and. P E PF. Describing P, given C, is entirely straightforward. For the other way around, micro-computer programs have recently been devised for drawing t9(P) for a given finite ordered set P.

17 One can finally establish that the dual of a finite distributive lattice is generally much smaller and less complex than the lattice itself. This just means that lattice problems concerning DF are likely to become simpler when translated into problems about PF.

Special properties of a finite distributive lattice are reflected in special properties of its dual. The following lemma which we just state without an actual proof (since the proof follows easily from earlier theorems) provides an elementary example.

Lemma 3.3.1 Let L = V(P) be a finite distributive lattice. Then

L is a Boolean lattice if and only if P is an antichain; 19(W) = 2n.

G is a chain if and only if P is a chain; 19(n).= n

Our next result serves a double purpose. It illustrates the way in which lattice constructs in DF correspond to ordered set constructs in PF. At the same time, it assists in the analysis and interpretation of the relation between complex lattices and their duals.

Theorem 3.3.2 Let P be a finite ordered set. Then:

19 (P)6 219 (P8); 19(P ® 1) o(P) ® 1 and 19(1 P) ^ 1 ® 19(P);

if P = P1 U P2 , where m and p2 are disjoint, then 19(P) 2-= 19(P1) x 19(P2)•

Note that, in (ii) above, ® denotes the linear sum of ordered sets, i.e: Let P and Q be (disjoint) ordered sets. The linear sum P ®Q is defined by taking the following order relation on P U Q: x < y if and only if x, y E P and x < y in P, or x,y E Q and x < y in Q, or x E P, y E Q.

Proof:

The required isomorphism is given by U P \ U for U E 19(P). The verification of this isomorphism follows easily.

The down - sets of P ED 1 are the down - sets of P together with P ® 1 itself. The down - sets of 1 ® P are the empty set and all down - sets of P with the least element of 1 ® P adjoined. The required isomorphisms are now easily set up.

18 (iii) It is easily verified that the map U H (U n P1 , U n P2 ), for U E 19(P), defines an isomorphism from 19(P1 U P2 ) to 79(P1 ) x 19(P2).

In the foregoing paragraph, we briefly introduced some correspondence between finite dis- tributive lattices and finite ordered sets. The importance of the duality in the finite case is clear: statements about finite distributive lattices can be translated into statements about finite ordered sets and vice versa. These ideas will be taken, with some helpful conclusions, further in the infinite case in the next chapter.

19 Chapter 4

Representation Theory Involving Boolean Algebras and Distributive Lattices : the General Case.

This chapter contains representation theorems for arbitrary Boolean algebras and bounded distributive lattices. Setting up the theory requires more sophisticated mathematics than in the finite case. Once obtained, however, the representation theorems are often as simple to use as their finite versions. We start this chapter with some background and definitions which will be needed in the later theorems.

4.1 Ideals and Filters

Definitions:

(i) Let be a lattice. A non-empty subset J of £ is called an ideal if:

a,bEJimplyaVbE a E b E „7" and a < b imply a E

(ii) A non-empty subset g of a lattice G is called a filter if:

a,bEgimplyaAbEg a E C, b E g and a > b imply a E g•

Note that an ideal or filter is called proper if it does not coincide with C. Also note that, for each a E .C, the set J. a is an ideal - the so - called principal ideal generated by a; dually, T a is a principal filter.

20 Let .0 be a lattice and J a proper ideal in C. Then is said to be prime if a, b E and a A be imply a E or b E 3. The set of prime ideals of G is denoted /p (r). A prime filter is defined dually and the set of prime filters is denoted by aTp(r).

Let G be a lattice and I a proper ideal in L. Then I is said to be a maximal ideal if the only ideal properly containing I is C. A maximal filter, more usually known as an ultrafilter, is defined dually.

Let S be a non-empty set. An ultrafilter of the Boolean lattice 2(8) is called an ultrafilter over b. An ultrafilter on b is said to be principal if it is a principal filter, and non - principal otherwise. The following two theorems enable us to have some deeper insight into the represen- tation theory involving arbitary Boolean algebras.

Theorem 4.1.1 . Let ,C be a distributive lattice with greatest element 1. Then every maximal ideal in .0 is prime. Dually, in a distributive lattice with least element 0, every ultrafilter is a prime filter:

Proof:

Let I be maximal ideal in G and let a, b E C. Assume a A b E I and a I; we require b E I. Define /a =1 {a V c I c I}. Then it is easy to check that /a is an ideal containing I and a. Because I is maximal, we have /a = C. In particular 1 E /a , so 1. = a Vd for some d E I. Then I (a A b) V d = (aV d)A(bV d)=bV d. Since b < bV d we have b E /.

Theorem 4.1.2 Let B be a Boolean lattice and let I be a proper ideal in B. Then the following are equivalent:

I is a maximal ideal;

I is a prime ideal;

for all a E B, it is the case that a E I if and only if ft I.

Proof:

(i) (ii): This follows directly from the foregoing theorem.

21 (iii): For any a E 5, we have a A = 0. Because I is prime, a E I or —.a E I. If both a and belong to I then 1 = a V E I - a contradiction.

(i): Let J be an ideal properly containing I. Fix a E J \ I. Then E I C J, so 1 = a V —.a E J. Therefore J = 5, which shows that I is maximal.

We just mention the following equivalences, which can be easily proved by using the last two theorems as well as the basic definitions.

Let .F be a proper filter in 2(.5). Then the following are equivalent:

(1) .T is an ultrafilter;

is a prime filter;

for each A C 5, either A E .T or 6. \ A E

a subset B of b belongs to .7- if A n B for all A E .F;

given pairwise disjoint sets A l., , A n such that A l U U An = 8, there exists a unique j such that Ai E

4.2 Representation by lattices of sets

The following two lemmas (of which we are only going to prove the second one) follow easily from the definitions, in connection of ideals and filters, introduced in the first few paragraphs of this chapter. It gives a clue on how to extend the representation theorems in chapter 3 and will help to show those results to be special cases of those we will obtain in this chapter.

Lemma 4.2.1 (i) Let ,C be a finite distributive lattice and a E L. Then the map x 1-4 C\ T x is an order - isomorphism of

{x E j(L) I x 5 a} onto {I E /4C) I a sl /1.

(ii) Let B be a finite Boolean algebra and let a E B. Then the map x H B\ T x is a bijection of {x E A(L) x < a} onto {I E /p (5) I a fl /}.

Lemma 4.2.2 Let r be a lattice and let X = Ip(r). Then the map y : G --+ P(X) defined by y : a X a := {/ E Ip (C) I a /} is a lattice homomorphism.

22 Proof:

We have to show that X avb = X a U X b and XaAb = Xa fl Xb, for all a, b Er. Take I E 4(L). Since I is an ideal, a V b E I if and only if a E I and b E I and, since I is prime, a A b E I if and only if a E / or b E I.

Thus we have

X avb = E -1-1,(L) I aVb0/} {/ E /p (C) I a 0 I or b O /} Xa U Xb•

Similarly, XaAb = Xa fl X b .

We should like the above map 71 to give a faithful copy of G in the lattice P(/ p (C)). We certainly cannot prove this without the additional hypothesis of distributivity, because a lattice of sets must be distributive. This problem is overcome by the following lemma, which shows exactly what is needed to ensure that a distributive lattice G has enough prime ideals for 77 : G --+ P(Ip(C)) to be an embedding.

Lemma 4.2.3 Let .0 be a lattice. Then the following are equivalent:

G is distributive;

given an ideal of G and a filter g of G with fl g = cb, there exists a prime ideal I such that J C I and I fl g 0;

.given a, b E G with a b , there exists a prime ideal I such that a 0 I and b E I;

the map 77 : a Xa := {I E Ip (r) I a 0 I} is an embedding of r into P(Ip (r));

.0 is isomorphic to a lattice of sets.

Proof:

Before we start with the actual proof, we first consider the following two statements which can be proved using the Axiom of Choice:

(DPI) Given a distributive lattice r and an ideal J and a filter g in .0 such that Jn g 0, there exist I E Ip(r) and F=L\IE ■Fp(L) such that J C I and g c F.

(BPI) Given a proper ideal J in a Boolean lattice B, there exists I E 4(8) such that C I.

Now we can start to prove the given statements of the lemma equivalent:

23 The implications (iv) (v) (i) are trivial and (i) (ii) is the statement that (DPI) holds. To prove (ii) (iii) just take =j, b and c =i a in (DPI).

Since 77 is a homomorphism, it is order - preserving. To prove that (iii) --+ (iv) it is enough to show that a b implies Xa Xb. This is true since the prime ideal I supplied by (iii) belongs to Xa \ Xb.

For Boolean algebras we have the following equivalent result which we just state here without a proof.

Lemma 4.2.4 Let B be a Boolean algebra. Then

(1) given a proper ideal J in B there exists a maximal ideal I E /7,(8) with J C I;

given a b in B, there exists a maximal ideal I E 4,(8) such that I contains one and only one of a and b;

there is a Boolean algebra isomorphism from B onto a subalgebra of P(X), where X = Ip(B).

In lemmas 4.2.3 and 4.2.4 above we have set out to prove analogues for theorems 3.2.2 and 3.1.2 of chapter 3, not restricted to finite lattices, respectively. We have in part achieved this by noticing the following two results (which follow easily from the work done in chapter 3):

Any distributive lattice is isomorphic to a lattice of sets.

Any Boolean algebra is isomorphic to an algebra of sets.

Nevertheless, what lemmas 4.2.3 and 4.2.4 look at is a characterisation of the image of the embedding i . We can however say straightaway that 7/ cannot always onto. Notice that every Boolean algebra is isomorphic to a powerset algebra and every distributive lattice is isomorphic to a lattice V(P). The description of im(77) has to be in terms of an additional structure on the set of prime ideals. A topological structure is exactly what we need. We therefore give a brief background on the most important definitions, concepts and results regarding a topological space, which we will need further on in this chapter.

Definitions:

(i) A topological space (X ,'T) consists of a set X and a family T of subsets of X such that E and X E T ; a finite intersection of members of T - is in T; an arbitary union of members of T is in T.

24 The family T is called a topology on X and the members of T are called open sets.

Given a topological space (X, T), we define a subset of X to be closed if it belongs to r(X):-.{.xvi U E T}.

Subsets of a topological space which are both open and closed are called clopen. Note that the family of clopen subsets of a topological space (X, T) forms a Boolean lattice. This suggests that, given a Boolean lattice B, we might try to impose a topology, T, on Ip (B) so that im(i) is characterized as the family of clopen subsets of (Ip(B),T). Define T on X := Ip (B) as follows:

T:={UCXIU is a union of members of im(y)

The topological space (Ip (B),T) is called the prime ideal space or dual space of B.

Let (X, T) be a topological space and let U := {Ui } iEi C T. The family U is called an open cover of Y C X if Y C U iE/ Ui . A finite subset of U whose union still contains Y is a finite subcover. We say Y is compact if every open cover of Y has a finite subcover.

Lemma 4.2.5 Let (X,T) be the prime ideal space of the Boolean lattice B. Then the clopen subsets of X are exactly the sets X a for a E B. Further, given distinct points x, y E X, there exists a clopen subset V of X such that x E V and y 0 V.

Proof:

Bearing the definition of a prime ideal space as well the elements contained in every X a in .mind, it is clear that each set X a is clopen. Also given distinct elements II and 12 of ip(8), there exists, without loss of generality, a E 1 .1 \ 12. Then Xa contains /2 but not I. This proves the final assertion.

It remains to prove that an arbitary clopen subset, U, of X is of the form Xa for some a E B. Because U is open, U = U.EAXa for some subset A of B. But U is also a closed subset of•X so that it follows from the definitions above that U is compact. Hence there exists a finite subset Al of A such that U = UaE,q , Xa. Then U = Xa , where a = .(>

We are now in the position to state an analogue for theorem 3.1.2 (chapter 3) which is not just restricted to finite lattices only.

Theorem 4.2.6 Stone's representation theorem for Boolean algebras

Let B be a Boolean algebra. Then the map

: a :={IE Ip (B) I a 0 I}

is a Boolean algebra isomorphism of B. onto the Boolean algebra of clopen subsets of the dual space (Ip (5),T) of B.

25 Proof:

The proof follows directly by combining lemma 4.2.4 as well as the first part of lemma 4.2.5.

We now return to distributive lattices and try, similar to that for Boolean algebras, to obtain an analogue for theorem 3.2.2 (chapter 3) which is not just restricted to finite lattices only.

Let L be a distributive lattice and let X = ip (C) be its set of prime ideals. We already have representations, following easily from theorem 3.2.2 as well as the work done so far in this chapter, for L in two special cases:

When L is Boolean and X is topologised in the natural way, L is isomorphic to the algebra 2(X) of clopen subsets of X.

When L is finite, L is isomorphic to the lattice V( X) of down - sets of X.

The above observations suggest that to represent L in general we should equip X with the inclusion order and a suitable Boolean space topology. A prime candidate for a lattice isomorphic to L would then be the lattice of all clopen downsets of X.

Let L be a distributive lattice with least element 0 and greatest element 1. Let X := I p (G). We want a topology, T, on X so that each Xa , a E L, is clopen. Accordingly, we want every element of S := {Xb I b E L} U {X \ X, I c E L} to be in T. Compared with the Boolean case we have a double complication to contend with. The family S contains sets of two types and it is also not closed under finite intersections. We let := {X b n c) I b, c E L} Since L has 0 and 1, 8 contains S. Also 8 is closed under finite intersections. Finally we define T as follows: U E T if U is a union of members of B. Then T is a topology on X - it is indeed the smallest topology on X containing S as well. The topological space (Ip (L),T), endowed with the inclusion order C, is called the prime ideal space or dual space of the bounded distributive lattice .C, and it is denoted by (ip (r), T, C).

The next lemma, very similar to lemma 4.2.5, characterizes clopen sets and clopen down - sets in the dual space (ip (L), T, C) of L. Since the proof follows the same lines as that of lemma 4.2.5 we just state the lemma without any formal proof.

Lemma 4.2.7 Let L be a bounded distributive lattice with dual space (x,r,c), where X = Ip (L). Then .

(1) the clopen subsets of X are the finite unions of sets of the form ..Kbn(X\X e) for b, c E L;

(ii) the clopen down - sets of X are exactly the sets Xa for a E L.

26 We are now in the position to state an analogue for theorem 3.2.2 (chapter 3) which is not just restricted to finite lattices only.

Theorem 4.2.8 Priestley's representation theorem for distributive lattices Let .0 be a bounded distributive lattice. Then the map

: a ► X a := {/ E ip(G) I a ¢I} is an isomorphism of .0 onto the lattice of clopen down - sets of the dual space (14C),'T,C) of G.

Proof:

The proof follows directly by combining lemma 4.2.3 as well as the second part of lemma 4.2.7.

4.3 Duality

In chapter 4, so far, we have presented Boolean algebra results first, and then their dis- tributive lattice counterparts. While we were setting up the necessary machinery this had advantages, since the Boolean case was somewhat simpler. Therefore we can place the em- phasis on distributive lattices, deriving Boolean algebra results as corollaries by specialising to the discrete order.

In chapter 3 we mentioned the following result: Suppose P is .a finite ordered set. Then the map e : is an order - isomorphism from P onto J(19(P)).

We are now going to discuss two theorems which will give us a clue on how to think in order to generalize the above result. It will also provide a way to obtain dual spaces regarding Boolean algebras and distributive lattices.

Definitions:

(i) Let (X, T) and (X', T') be topological spaces and f : X —4 X' a map. Then the following conditions can be proved to be equivalent:

(i) f -1 (U) is open in X whenever U is open in X'; f -1 (V) is closed in X whenever V is closed in X';

27 (iii) f -1 (U) is open in X for every U E S, where S is a given basis or sub-basis for T'(it being understood that a basis S for T' is a specified family of open sub-sets of S which are closed under finite intersections. However, if S is not closed under finite intersections, we first form B, the family of sets which are finite intersections of members of S, and then define T to be all arbitrary unions of members of B. In this case S is called a sub-basis for T.

When f satisfies any oof the above 3 conditions it is said to be continuous.

Let (X, T) and (X', T') be toplogical spaces and f : X ---÷ X' a map. Then the map f is said to be a homeomorphism if it is bijective and both f and f-1 are continuous. Homeomorphisms are therefore topology's isomorphisms.

A map f : X X' from an ordered space X to an ordered space X', which is simulta- neously an order - isomorphism and a homeomorphism is called an order - homeomorphism. X and X' in this case are said to be order - homeomorphic.

A set X carrying a topology, T, and an order relation, <, is called an ordered (topological) space and denoted (X, T,<). It is said to be totally order - disconnected if, given x, y E X with x X y, there exists a. clopen down - set U such that x E U and y 0 U. We call a compact totally order - disconnected space a CTOD space. (CTOD spaces are also known as ordered Stone spaces or Priestley spaces).

Theorem 4.3.1 (i) Let Y be a Boolean space, let B be the algebra P(Y) of clopen subsets of Y and let X be the dual space of B. Then Y and X are homeomorphic.

(ii) Let Y be a CTOD space. Let C be the lattice 19(Y)• of clopen down - sets of Y and let X be the dual space of G. Then Y and X are order - homeomorphic.

Proof:

Note that part (i) is essentially the special case of part (ii). Therefore we just prove (ii):

We define e : Y --+ X by c(y) := E G I y0 a). Certainly e(y) is a prime ideal in G. We shall show that

e is an order - embedding and therefore one - to - one;

e is continuous;

c maps Y onto X.

98

To prove that e is an order - embedding we just have to note that

y < z in Y < > (VaEG)(zEayEa) < > e(y) C e(z).

It is now easy to show that e is one - to- one as well. Therefore (a) is established.

In order to prove (b), we notice that, according to the definition of a continuous mapping, (b) holds so long as E-1 (Xa ) and E-1 (X \ Xa ) are open for each a E C. But

E-1 (X \ Xa ) = {y E Y E(y) 5tXa} = Y \ E -1 (Xa )•

Thus (b) holds provided E -1 (Xa ) is clopen in Y for each a E C. But

E-1 (Xa) {y E I c(y) E Xa } = {yEY1a0 f(Y)}... by def. of Xa a ... by def. of e,

and this is clopen, by definition of C.

Finally we prove (c). It can be proved easily that e(Y) is a closed sub-set of X. Suppose by way of contradiction that there exists x E X \ e(Y). This implies that there is a clopen sub-set V of X such that E(Y) fl V = 15 and x E V. By lemma 4.2.7, we may assume that V = X b n .x,) for some b, c E G We have = Y n c 1 (V) = b n \ c). Thus b C c, which is impossible since x E Xb fl (X \ -3(c ) - a contradiction.

Now, since (a), (b) and (c) have been established, it follows easily from the properties regarding topological spaces that (ii) holds.

It is far from easy to find all the prime ideals of an infinite lattice and describe the structure of the dual space. Fortunately the following corollary to the foregoing theorem provides an indirect way to obtain dual spaces regarding Boolean algebras and distributive lattices. Since the theorem follows trivially from theorem 4.3.1 we just state it without any proof.

Theorem 4.3.2 (i) Let B be a Boolean algebra andY a Boolean space such that P(Y) B. Then the dual space of B is (homeomorphic to) Y.

(ii) Let C be a bounded distributive lattice and Y a CTOD space such that 19(Y) 1 7-.' L. Then the dual space of C is (order - homeomorphic) to Y.

29 Chapter 5

Representation Theory: A more general and applicable view.

In chapter 3 and 4 we introduced representation theorems regarding Boolean algebras and distributive lattices. In chapter 5 these ideas will be developed further in a so - called more general way. More representation theorems in some other universal genres will be studied. Most of the proofs in paragraphs 5.1 and 5.2 follow directly from B.A. Davey and H.A. Priestley: "Introduction to Lattices and Order". Proofs in later paragraphs of this chapter follow from the following articles:

"The Theory of Binary Relations" - Bjarni JOnsson

"Introductory Course on Relation Algebras, Finite - Dimensional Cylindric Algebras, and their interconnections" - R.oger D. Maddux

"The Representation of 3 - dimensional Cylindric Algebras" - Stephen D. Corner

5.1 Analogue statements for the representation the- orems of chapters 3 and 4

Although considered from a general viewpoint, the representation theorems regarding Boolean algebras and distributive lattices introduced in chapters 3 and 4 were "overflowed" with some specific (and non-general) . notation. We therefore start this chapter with some equivalent and more precise statements.

The following theorem is a generalization of Birkhoff's representation theorem for finite distributive lattices. It can also be seen as a generalization of Priestley's representation theorem for distributive lattices.

30 Theorem 5.1.1 Every distributive lattice (X, A, V) is isomorphic to a sublattice of a set lattice (P(F(X)),n,u) for some set F(X).

Proof:

Let F(X) be the set of all prime filters in X. Define a mapping h : X P(F(X)) as follows: h(s) = { ik E F(X) I x E 0}- h is an isomorphic embedding:

h is a homomorphism: One can easily show that h(x A y) = h(s) fl h(y) and h(x V y) = h(x) U h(y) so that h is a homomorphism.

h is injective: If x• y then either •a: y and y x. But one can easily show (by using Zorn's lemma) that, if (X, A, V) is a distributive lattice and x i , yl E X are such that x i yi then there is a prime filter q5 in X such that x i E ck and yi 0 0. Therefore it follows that h(x) # h(y), so that h is injective.

Thus, (X, A, V) is isomorphic to (8(X), A, V) where

8(X) = {h(x) I x E X}.*

Note that, in the above theorem, the mapping h, the set F(X) and the lattice S(X) are called respectively the Stone isomorphism, the Stone space and the Stone lattice of the lattice X.

Mathematicians often prefer to use this version of the representation theorems for distributive lattices. This is partly because the theoreM is more applicable under general circumstances and notationalwise less involved.

Before going any further with analogue theorems for the representation theorems involv- ing disrtibutive lattices and Boolean algebras, we need, apart from the definitions given in chapter 4, some more background regarding topological spaces.

Definitions:

The topological space (X, T) is said to be Hausdorff if, given x, y E X with x # y, there exist open sets U1 , U2 such that x E U1 , y E U2 and U1 fl U2 = •

A topological space (X, T) is said to be a To-space if, for every x, y E X, there exists an element U of T such that either x E U and y U or x U and y E U.

31 The following 3 representation theorems are all extensions of theorem 5.1.1. and are proved analoguously. We, therefore, just state the theorems here without any proofs. Note the equivalence between the third theorem and Stone's representation theorem for Boolean al- gebras (chapter 4).

Theorem 5.1.2 (Representation theorem for implicative lattices - Rasiowa and Sikorski : 1953) Every implicative lattice without a bottom (element) 1 is isomorphically embeddable into the lattice of open and dense(i.e.: all subsets which meet with all other non-empty open subsets) subsets of a compact topological To -space.

Theorem 5.1.3 (Representation theorem for Heyting algebras - Stone : 1937) Every Heyting algebra is isomorphically embeddable into the algebra of open subsets of a compact topological To -space.

Theorem 5.1.4 (Generalization of Stone's representation theorem for Boolean algebras)

Every Boolean algebra is isomorphic to the algebra of open and closed subsets of a compact, totally disconnected topological space. A simplified version of the above statement is the following: Every Boolean algebra is iso- morphically embeddable in the algebra of subsets of some set.

Note that, like in the case of distributive lattices, theorem 5.1.4 is more applicable under general circumstances and notationalwise less involved than the version in chapter 4. Math- ematicians therefore prefer theorem 5.1.4 in general representations and applications, as we will see later on.

5.2 Introduction to relation algebras, finite - dimen- sional cylindric algebras, and their interconnec- tions

Up to here we described the notions of algebras, Boolean algebras and equivalent concepts. Some scripts might introduce the background regarding two further structure concepts, namely relation algebras and cylindric algebras, in a separate chapter, but because of the

32 fact that these structures are equivalent to structures introduced earlier on, we just consider a brief introduction to these structures in this paragraph.

Relation algebras are closely linked to cylindric algebras, and yet in certain ways they are quite different. The links and differences allow for constructive interaction between these objects. Relation algebras arise naturally by considering families of binary relations and certain operations on binary relations. We start by giving a definition of Boolean algebras, which is equivalent to the definition given in chapter 2. The definition which we give here was formalized by De Morgan in 1856:

5.2.1 Formalized definition of a Boolean algebra

Boolean algebras may be considered as algebras whose elements are sets, and whose opera- tions are certain operations on sets. For any two sets a and b, let

a - b = {x I sea or

a • b fa: I x E a and sx bb}} 1 a® b = {x I (a: E a and x b) or (x 0 a and x E b)} .

Thus a + b is the union of a and b, a • b is the intersection of a and b, and a e b is the symmetric difference of a and b. The empty set is 0. With respect to some fixed largest set U, denoted by 1, the complement Zi of a is defined by

= {s I se u and x 0 a}.

Thus xey=x-V-F T: y and 0 = T.

Now we are in the position to state a formalized definition of Boolean algebras:

A Boolean algebra of sets is an algebra of the form (A, -F, —), in which A is a nonempty family of subsets of some set U and A is closed under union (-1-) and complementation (—) with respect to U. A Boolean. algebra is any algebra which is isomorphic to a Boolean algebra of sets.

5.2.2 Further definitions

Proper relation algebras are certain Boolean algebras of sets whose elements are binary relations, and whose operations are certain operations on binary relations. For any two binary relations a and b, let

a; b := {(x, z) (ay)((x,y) E a and (y, z) E b)}, a:=. {(y,x) (x,y) E a ) .

33 Thus a; b is the relative product (or composite) of a and b, it is the converse of a. The operation ; is relative multiplication (or composition), and the operation is conversion. If U = {x (3y)((x,y) E e and (y,x) E e)}), then

1' = {(x,y) x EU}, 0' = {(x,y) I x, y EU and x y} = 1 1 , 1 = 1' + 0', 0 = I".

1' is the identity relation, 0' is the diversity relation, 1 is the universal relation, and 0 is the empty relation.

We are now ready to state the following definitions:

A proper relation algebra is an algebra of the form (A,+,—,•,U, 1'), in which A is a nonempty family of binary relations contained in some largest relation e with field U,1' E A, and A is closed under union, complementation with respect to e, relative multiplication, and conversion.

A relation algebra is an algebra (of the same form as a proper relation algebra), but with the extra condition namely that it satisfies a finite set of identities which hold in all proper relation algebras. In his article, "Inroductory course on Relation Algebras, Finite - Dimensional Cylindric Algebras, and their interconnections", Roger D. Maddux gives a detailed discussion of these identities.

A relation algebra U is representable if it is isomorphic to a proper relation algebra. A representation of U is an embedding of U into a proper relation algebra. The main purpose of cylindric algebras is that it provide an algebraic semantics for first order(predicate) logic. One could say more about relation structures by means of cylindric properties. The construction of cylindric algebras from relation algebras won't be studied in this thesis. For further reading, the article by Roger D. Maddux, "Introductory course on Relation Algebras, Finite - Dimensional Cylindric Algebras, and their interconnections", is recommended. A • somewhat detailed discussion of the simpler algebraization of classical propositional logic by Boolean algebras occurs at the end of chapter 7. Generalized cylindric set algebras of dimension a are certain Boolean algebras of sets whose elements are a-ary relations over a set U, and whose operations are certain operations on a-ary relations. In this case the largest relation has the form e aui , where (Ui , i E I) is an I-indexed sequence of pairwise disjoint sets, and U = (au, is the a-th direct power of i.e., au, = x x (a-times), and Eiv U1 is the union of the Ui 's). The Boolean operations are union and complementation with respect to e. For any a-ary relation a C e and k < a, let • cka = fs x E e and (3y• E a)(V) < a)(A k sA WI • We say cka is the loth cylindrification of a (with respect to e). Also, for any k, A < a,

34 • do, ={xI xEe and x k = x ),}. The relations do, are called diagonal relations in e. We are now ready to state the following definitions.

An a-dimensional generalized cylindric set algebra is an algebra of the form Lf = (A,+,—,ck ,dkA) k ,),<,,, in which A is a non - empty family of relations on a set U, called the base of A; A has a largest relation e = E1E1 aui , where (U1 , i E I) is a partition of U; A contains the diagonal relations in e and A is closed under union, complementation with respect to e and all cylindrificatioris with respect to e. U is a cylindric set algebra if e = cxu .

A cylindric algebra U is representable if it is isomorphic to a generalized • cylindric set algebra. A representation of U is an embedding of U into a generalized cylindric set algebra.

We introduced more involved concepts in the foregoing paragraphs. In the next paragraphs, we will study the role which these concepts play in general representation theory. It will also become clear that the representation theorems which we will study in this paragraph result in "round - off" - and application theorems of these studied in chapters 3 and 4.

5.3 Representation of Relation Algebras

The whole process of representing a relation algebra is very similar to the idea of representing pure Boolean algebras. However, as defined earlier, one must bear in mind that a relation algebra is a proper relation algebra. which satisfies a finite set of identities which hold in all proper relation algebras.Because of the fact that the representation theorems which hold for pure Boolean algebras also hold for relation algebras(sometimes with some more involved conditions), we are not•going to discuss the representation of a relation algebra in that detail. We rather pay more attention to 3 basic facts about the representability of relation algebras which play a very important role when representing ralation algebras:

The class of all representable relation algebras (denoted by RRA) is not equal to the class of all relation algebras (RA) . . . Lyndon : 1950.

RRA is an equational class ( or a variety), where a variety can be defined as a class axiomatized by a set of identities ... Tarski : 1955.

RRA is not finitely based (i.e.: there is no finite set of identities that axiomatizes RRA ) ...Monk : 1964.

The proofs of the above 3 statements are very algebraic since they make use of a lot of general facts from universal algebra. Also note that the theorem is a sophisticated model theoretic tool for the study of first order logic - therefore it must been compared with a

35 more detailed treatment of the simpler study of propositional logic at the end of the thesis. We start by proving the second statement which proclaims that RRA is an equational class:

Denote by Var(k) the variety generated by a class 1C of algebras. Let P, S, H, Ps and Pu denote the operations of forming direct products, subalgebras, homomorphic images, subdirect products and ultraproducts. Let

= {B B R(X) for some set X}, where B is a non-trivial algebra and R(X) is the full algebra of binary relations on X. Then RRA = SP(), and hence, since we know from Birkhoff's Theorem that Var(1C) = HSP() for any class K of similar algebras, Var(RRA) = Var(k). Since RA is congruence distribu- tive(see Bjarni kinsson's article on the theory of binary relations for a detailed discussion on this), it follows that

Var(RRA) = PsHSPu(k) = SPHPu(k) = SPPu(K).

Note that the above conclusion made use of two quite deep results, namely JOnsson's Lemma which states that the subdirectly irreducible algebras in a congruence distributive variety Var(k) lie in HSPu(k) and Birkhoff's subdirect decomposition theorem(which is discussed in chapter 7). In the last step we have used the fact that every member of K is simple (which means that, in the scope of the present work, every member of k is subdirectly irreducible and directly indecomposable) and that Pu preserves simplicity (bearing in mind that relation algebras form a discriminator variety - see "A course in Universal Algebra" by Burris and Sankappanavar). Now RRA is closed under P and S. To complete the proof, it therefore suffices to show that Pu(K.) C RRA. To prove this inclusion, we consider the class. G of all atomic relation algebras B - i.e.: algebras with the property that, for all atoms x, y E B, x; 1; y is either an atom or the zero element. Certainly, from Los'Theorem, it follows that C .1: and Pu(L) = ,C because G is an elementary class(i.e.:a class of elementary structures where one structure is either an elementary substructure or an elementary extension of another). To show that ./2. C RRA, consider any B E ,C and let U be the atomic structure of B(i.e.:14 contains and forms a basis for all the atoms ofB . Then B is isomorphic to a subalgebra of Cm(U) (where Cm(U) = (B(U), R o , with B(U) the Boolean algebra of all subsets of U and, for X = (X0, X1, , Xn _1) a sequence of subsets of U, Ri(X) is the union of all the sets R=(x) with x E X0 x x x Xn_ 1 .

R=(x) = {y E U I (x0,•x1, • • • Xn-11 y) E R} with Ri a relation of rank n 1 on U). One can easily prove that Cm(U) is representable, so that it follows that Pu(k) C Pu(G) = G C RRA. 0

We omit the proofs of (i) and (iii) because it would require a detour into methods beyond' the scope of the present work. In chapter 6, where Boolean algebras with operators will be studied in more detail, we will give more insight to the problem of representing Relation algebras.

36 5.4. Representation of finite - dimensional cylindric algebras

When the representation of a cylindric algebra (denoted by CA) is mentioned, one normally thinks about representing the algebra as a subdirect product of cylindric set algebras. The class of C A,'s (cylindric algebras of dimension a) having such a (set) representation is denoted by RCA,. Like in the case of relation algebras, this representation is very similar to earlier results, so that we rather consider some more involved constructions which also play an import role. when representing CA's. For the sake of simplicity, we restrict ourselves to cylindric algebras of dimension 3 in this paragraph. For further reading, the article by Stephen D. Comen, "The Representation of 3 - dimensional Cylindric Algebras", is highly recommended.

Note that the following definitions, as well as the theorems which will be discussed further-on in this chapter, are very general. Therefore, a good understanding ability(as some of the definitions will not be defined in that detail) on the articles mentioned at the beginning of this chapter, are highly recommended for the uninitiated reader.

Definitions:

(i) A cylindric algebra of dimension 3, or CA 3 , is a system U = (A, - F, • , —, 0, 1, ci , dii)1j.<3 , ci 's unary and dij nullary operations, that satisfies the following conditions for all j, k < 3 and all x,y E A: (C 0) (A, +, •, —, 0, 1) is a Boolean algebra, ci0 = 0, x < cix, ci (x • ciy) = cis • ciy, cicix = ci cix, dii = 1, ci (di; • dik ) = dik if j k, ci (di; • x) • ci (d.i; • —x) = 0 if

(ii) A CA3 is called complete and atomic if its Boolean part is complete and atomic. From a structure B = (B, T , B x B and B, we obtain a complete atomic CA 3 as follows:

C,n 8 = (SbB, U, n, ti , q5, B, ci , Eii ) i,;<3 where, for X C B and i < 3,

c,X = {y E B xTiy for some x E X} .

Cre.,B is called the complex algebra of B.

37 Note that every relation or cylindric algebra IA has an atom structure WU. Complete and atomic CA's can be studied by structures on the atoms. The cylindric atom structure of a complete and atomic CA 38 is a system UtU = (At13, Eii)i,j <3 , where

Ti = {(x, y) E ALB x AtB c ix = ciy}

and Ei; = Ix E At5 x < dii}.

(iii) The cylindric atom structure of a complete atomic CA 3 can be characterized as a

relational system 13 = (B, Ti , Ei; ) 1 ,j<3 where Ti C B x B, Ei; C B such that for all i,j, k < 3:

Ti is an equivalence relation on B, TA = Eii = B, Eij = Tk(Eik n Eki) for i,j k, ITix n Eiji! = 1 whenever x E B, i j.

(iv) For cylindric atom structures B = (B, Ti , Eii ) jj<3 and B' (B', j) 1 j<3 a function h from B to B' is a full homomorphism of B onto B', in symbols h : B B', if, for all x,y E B and i, j < 3, ay < > (hx)T((hy) and x E < > (hx) E E.

(v) For complete and atomic CA 3 's U and B, U is an adjunction of B, in symbols U E Adj(B), if h : LftLf >- MB for some full homomorphism h. If K is a class of CA3 's, let Adj(K) = U{Adj(B) I B E K).

(vi) A partial multi - valued loop is a structure M = (111,0,E) where Onot = E C M, a o b C M for all a, b E Al, and the following properties hold for all x, y E M:

there exists a unique e E E such that x o e= {x} and a unique f E E such that f o x = {x}. We denote e = r(x) and f = d(x). •

e o e = {e} for all e E E,

x o y iff 7-(x) = d(y),

xEyoz for some z E M iff d(x) = 7-(y),

xEzoy for some z E M iff r(x) = r(Y), there exists a unique z E M and a unique w E M such that d(y) E y oz and r(y) E woy.

The following theorems and lemmas play (directly or indirectly) an important role in the representation of CA's as different mathematical models.

38 Lemma 5.4.1 In a non-trivial CA3 U the following are equivalent:

Vx(x 0 coci x = co c2x = ci c2x = 1),

d01 • d1 2 is an atom.

Proof: (ii): Suppose (i) holds and assume x,y 0, x • y = 0 and x y -= do 1 • d12 in U. Notice that cox + coy -=d12. From this and property (C 7) we can conclude that ci cox • cicoy = 0. So, not both coci x and cociy can equal 1 which contradicts (1). Thus, either d01 • d12 is an atom or d01 d12 = 0. But d01 • d12 = 0 implies U has only one element, so that (ii) holds.

(ii) (i): If coci x • LO1/ d12 = 0, then 0 = coc1 0 = co c1 (co ci x • do1 • d12) = ci (coci x • d12) = cocix which implies x = 0. Thus, if (ii) holds and x 0, then d01 • d12 < coci x which yields 1 = coc1(c/01 • d12) = coci x. The other cases are similar. •(>

A CA3 that satisfies property (i) above is called integral. Now, from lemma 5.4.1. above we can conclude that every (integral) CA 3 can be embedded in an (integral) CA 3 that is complete and atomic.

Lemma 5.4.2 (i) If 8 and B' are cylindric atom structures and B B', then C,„8' is embeddable in

ForU and t3 complete atomic CA 3 's, Lf E Adj(8) implies B is isomorphic to a subalgebra of U.

Proof:

If B B', there exists a. full homomorphism h of 8 onto B'. It is straight forward to check that the map x I a E x} is an embedding of Cm B' into Cm 13.

One can easily show that for every complete atomic CA 3 , U, if C m UtU. From this and (i) above we obtain (ii) directly. <>

For complete atomic CA 3 's U and B, we say that 13 is minimal with respect to U E Adj(B) if C E Adj(B) for every complete atomic C with U E Adj(C). In particular, it follows that Lf E Adj(.8).

39 If 0 is an equivalence relation on MU, we let Uttil0 denote the quotient of the relational system Utti by 0; Utt110 is the system (AtU10 , T1, g i) 1,;<3 where Attil0 = {Oxix E Atli} is the collection of all 0-blocks, (0x)T1(0y) <=> xTiy, and

Ox E <. > x E Ei; for all i,j < 3.

We are now ready to state and prove the following theorem.

Theorem 5.4.3 Suppose U is a complete atomic C A 3 with cylindric atom structure

Uttl = (Atli ,Ti, Eii)i,j<3 and 0= To nTi n 712. Then

Utt110 is a cylindric atom structure;

Cm Ut/410 is isomorphic to a subalgebra B oft' that is minimal with respect to U E Adj(B);. The subalgebra B of U in (ii) is unique. In fact it is characterized by

x e B if Va E Atti(a < x coa•ci a • c2a < x).

Proof:

(i) Clearly 0 is an equivalence relation on Atli such that for all x, y E Atli

Ay if cix = ciy for all i < 3.

Properties (1) - (3) of a cylindric atom structure are obvious for UtUIO. For the fourth property, note that

Ox E < > x E Ei; < > xTky for some y E Eikn Ejk

(Ox)T/(Oy) for some Oy G EL n Ekj

< > Ox E Ti:(EL n Eki ).

To show that the fifth property holds, suppose Ox, Oy E gi with i j and (0x)27(0y). Then x, y E Eii and xTiy so x = y (by the fifth property of a cylindric atom structure UtU). This finishes the proof of (i).

40

The map 0* : x H Ox shows that UtU >- Uttil0. Thus, by (i) of lemma 5.4.2., C,,littil0 is isomorphic to a subalgebra B of U. Now, if U E Adj (C) for some complete atomic C,

there is a full homomorphism h : UtC. For x, y E AtU , hx = by implies hxTi'hy and hence xTiy for all i. Thus, kOh C 0. Hence there exists a full homomorphism k : UtB litlijO such that 0* = k o h. Therefore, C E Adj(C„,Utti10). Thus, Cnititt110 is minimal with the property.

Suppose 13 is the subalgebra of U constructed in (ii). Since UtB litU10, the atoms of B are elements of the form E Ox where x E Atx and 0 = To fl Ti fl T2. Then

cox•c i x•c2x = co(E ox) • ci ( E ox) • c2 (E ox) E Ox for every x E Atli N

First note that ci(E Ox). = E ci Ox = cix for all i. Now, suppose x, y E Atli and y < cox • cix • c2x. Then ciy < cix and since cix is an atom of the BA of ci-closed elements of U,ciy = cix. Hence, x0y and y < E Ox. Thus, cox ci x • c2 x < E Ox which completes the proof of (i). Now suppose x E B, x 0, and a E Atli such that a < x. Then x > E Oct = coa • ci a • c2 a, the smallest atom of B that contains a. On the other hand, suppose x satisfies the condition U is atomic, so

x E{a E Atti I a x} = oa I a E AtU, a < x} E B

by and the condition on x. This completes the proof of (iii). •O• From the foregoing theorems and lemmas one can conclude (with a straight-forward proof, which we are not going to discuss) that every CA 3 can be represented as an adjunction of the complex algebra of a partial multi-valued loop. In this chapter we went into some more specific detail regarding the representation of structures introduced in chapters 3 and 4. Nevertheless, some of the ideas introduced in chapter 5 will be studied in even some more specific detail in chapter 6, where we will take a look at specific Boolean algebras with operators (of which relation algebras and cylindric algebras are examples).

41 Chapter 6

Boolean Algebras with Operators

The important role which representation theorems play when representing a abstract math- ematical structure as a less - involved and(sometimes) more general system, is clear up to now. However, it would not be complete without a bit more insight regarding arbitrary Boolean algebras with operators. We therefore dedicate this chapter to a short and general discussion on Boolean algebras with operators.

Since Stone and Birkhoff made a major breakthrough regarding representation theorems in connection with distributive lattices and Boolean algebras, various new kinds of al- gebraic systems have been defined and studied which can be referred to collectively as Boolean algebras with operators. These new algebraic systems have been obtained by enrich- ing Boolean algebras by means of some new operations which are assumed to be additive, i.e., distributive under ordinary Boolean addition. As examples relation and cylindric algebras (as discussed in chapter 5), cloSure algebras and projective algebras may be mentioned. Like most other algebraic systems which have been studied in modern algebra, all these systems have arisen as results of generalizations of specific models studied in various parts of Mathe- matics. Whenever a new kind of algebraic system originates in this way, the problem presents itself whether the abstract characterization is adequate in the sense that every system under discussion is isomorphic to one of the original models; this is the so - called representation problem for the given class of algebraic systems. As is well known, this problem has been completely solved for Boolean algebras without operators by showing that every Boolean algebra is isomorphic to an algebra formed by a field of sets under the set - theoretical oper- ations of addition (formation of unions) and multiplication (formation of intersections) (this was proved in chapters 3 and 4). It is also well known that this representation theorem is a simple consequence of (and trivially equivalently to) the so - called extension theorem by which every Boolea.n algebra can be extended to a complete and atomic Boolean algebra. The aim of the rest of this chapter is to briefly discuss the latter extension theorem as well as the equivalent representation theorem for Boolean algebras with operators. We will then take a very brief look on the application of these results to closure algebras and cylindric algebras. Most of the proofs in this chapter follow (directly or indirectly) from the articles "Boolean Algebras with Operators - Part (i) and (ii)", by Bja.rni kinsson and Alfred Tarski.

42 6.1 Definitions and Notation

Note that, in this chapter, we omit, for the case of simplicity, the complementation operation for Boolean algebras. However, it is worthwile to bear in mind that the complementation operation operates in very much the same way than the other operations consider.

(i) Let U = (A,+,0,•, 1) and B = (B, +, 0, •, 1) be two Boolean algebras. We say that 5 is a regular subalgebra of U and that U is a perfect extension of B if the following conditions are satisfied:

II is complete and atomic, and B is a subalgebra of U. If I is an arbitrary set, and if the elements xi E B with i E I are such that Eiv x i = 1, then there exists a finite subset J of I such that EiE, x i = 1. If it and v are distinct atoms of U, then there exists an element b E B such that < b and v • b= 0.

(ii) Let (A, +, 0, •1) be a fixed Boolean algebra. A function f from Am to A is called

normal if, given any j < in and a sequence x E Am such that x; = 0, we always have f(x) = 0; monotonic if, given two sequences x,y E Am such that x < y, we always have f(s) < 1(0; additive if, given any j < in and two sequences x, y E Am such that xp = yp whenever j p < in, we always have f(x y) = f (x) f(y); completely additive if, given any j

f(Ex (0 ) f(X (2) ). iEI

(iii) By a Boolean algebra. with operators we shall mean an algebra U = (A, +, 0, •, fo...) such that (A, +, 0, •1) is a Boolean algebra and the functions A are additive.

(iv) Suppose U is a Boolean algebra with operators. By an atom of U we mean an atom of the Boolean algebra (A, +, 0, 1). We say that U is atomic if the Boolean algebra (A, +, 0, •, 1) is atomic. We call IA complete if the Boolean algebra (A, +, 0, •, 1) is complete and if each of the operations A is completely additive. Finally, we say that U is normal if each of the operations A is normal.

(v) Let /4 = (A, +, 0, 1, , f4 , ...) and B = (B,-F,0,•,1, , ...) be two Boolean algebras with operators. We say that U is a perfect extension of B and that S is a regular subalgebra of U if the following conditions are satisfied:

U is complete and atomic and B is a subalgebra of U. (B, +, 0,•1) is a regular subalgebra of (A, +, 0, •1).

43 (c) For each of the functions ft we have ft (x) = nx

To have a better understanding of some of the concepts which are going to be defined in the next paragraphs, further reading from the articles "Boolean Algebras with Operators - Part (i) and (ii)", by Bjarni Jonsson and Alfred Tarski, is recommended.

6.2 The so-called extension theorem for Boolean al- gebras with operators

Before stating the actual extension theorem we give some necessary theorems and conditions.

Theorem 6.2.1 Let B = (A, +, 0, 1) be an arbitrary Boolean algebra. Then there exists a complete and atomic Boolean algebra U which is a perfect extension of B.

The above theorem is al so. referred to as the so-called extension theorem for Boolean algebras. As mentioned earlier on, this theorem is well-known. It can easily be proved by using basic definitions introduced in this chapter as well as some of the techniques used in chapters 3 and 4.

Through-out the next definition as well as the following short lemmas we shall consider a fixed Boolean algebra U = (A, -F, 0, •, 1) which is complete and atomic, and a regular subalgebra B = (B,-1-, 0, 1) of U. Note that an element x of A is said to be open if x = Es>yEB y and closed if x = 11..x

Definition:

For any function f on Bin to B, f+ is the function on Am to A defined by the formula f +(x) E f(z)) for any x E A', where C is the set of all closed elements of A. x

The proofs of the next three lemmas are quite similar and very technical and therefore we just state the lemmas Without any proofs.

Lemma 6.2.2 If f is a function on Br' to B, then f+(y) = ll y

44 Lemma 6.2.3 If f is an additive (or, more generally, a monotonic) function on Bm to B, then f+IBm = f.

Lemma 6.2.4 If f is an additive function on Bm to B, then f+ is a completely additive function on Am to A.

We now state and. prove the actual extension theorem for Boolean algebras with operators:

Theorem 6.2.5 Extension theorem for Boolean algebras with operators For any Boolean algebra with operators B there exists a complete and atomic Boolean algebra with operators U which is a perfect extension of B.

Proof:

Let B = (B,+, 0,•, -F, go,gi ,...,gc,...) be a Boolean algebra with operators. By theorem 6.2.1. the Boolean algebra (B,+,0,•,1) is a. regular subalgebra of a complete and atomic Boolean algebra (A, +, 0, •, 1). If we put A = gt for each 6, then, by lemma 6.2.3, B is a subalgebra of the algebra U = (A,+, Moreover, by lemma 6.2.4 and the involved definitions (see paragraph 6.1. (iii) and (iv)), if is a complete and atomic Boolean algebra with operators. Finally, the third condition of the definition of a perfect extension of a Boolean algebra with operators is satisfied by lemma 6.2.2.] Hence U is a perfect extension of B. <>

6.3 A Representation Theorem for Boolean Algebras with Operators

In the very first paragraphs of this chapter we mentioned the fact that the well-known representation theorem for Boolean algebras (as discussed in chapters 3 and 4) can be seen as an equivalent formulation of the so-called extension theorem for Boolean algebras. We are now going to prove an analogue theorem for Boolean algebras with operators by using the extension theorem for Boolean algebras with operators. We first introduce a few new concepts and notation:

Definitions:

(i) The system U = (A, Ro , R i , , Re ,...), where A is a non-empty set and each R c is a subrelation of Amcf l for some mo is called an relational structure.

45 A set field(i.e.: a set consisting of all open and closed sets in a totally-disconnected compact space) B whose universal set is U will be called regular if the Boolean algebra (B, U, 0, fl, U) is a regular subalgebra of the Boolean algebra (A, U, 0,11, U), where A is the family of all subsets of U.

Let B be a regular set - field, with U the universal set of B and A the family of all subsets of B. If R C Urn+1 , then the function R* on Am to A is defined by the formula

R*(X) = {y E U I (x0,...,x m _i ,y) E R for some

xo E X0, , x rn_i E for any Xi E Am.

By the complex algebra of a relational structure U = (U, Ro , R 1 , . . , Re , . . .) we mean the algebra U' = (A, U, 0, fl, U, , .) where A is the family of all subsets of U.

Lemma 6.3.1 If R C Um+1 , then R* is a normal and completely additive function on A m to A where B is a fixed regular set - field with U the universal set of B and A the family of all subsets of U. Conversely, if F is a normal and completely additive function on Am to A, then there exists a unique relation R C Um+1 such that F = R*. This relation is defined by the formula

R = {x o y Ix= (xo , x i , ... ,x„,_1 ) E Um and y E Fftro b {x i }, ...,{xm_ i })].

Since the proof of the above theorem requires some involved techniques, which are beyond the scope of the present work, we omit the proof.

Lemma 6.3.2 The complex algebra of any relational structure is a normal, complete, and atomic Boolean algebra with operators. Conversely, every normal, complete, and atomic Boolean algebra with operators is isomorphic to the complex algebra of some relational structure.

Proof:

This part follows easily from definition 6.1. (iii) and (iv) as well as lemma 6.3.1 and the definition of a complex algebra of a relational stucture.

Suppose U = (A, +, 0, 1, fo , , .) is a normal, complete, and atomic Boolean algebra with operators. Let the rank of A be n24. . Then U is isomorphic to an algebra = (A, U, 0, fl, U, Fo, F1, . , Fe, . . .) where U is some set, A is the family of all subsets of U, and Fe is a normal and completely additive function on Ante to A. By lemma 6.3.1 there exists relations Ro , R1, . , Re, . . . such that Re C Umc+ 1 and Fe = RZ. Hence; by the

46 definition of a complex algebra of an relational structure, U' is the complex algebra of the

relational structure U" = (U, Ro , R 1 , . . . , Re , . . .).

We are now in the position to state and prove a representation for Boolean algebras with operators.

Theorem 6.3.3 (Representation theorem for Boolean algebras with operators) Every normal Boolean algebra with operators is isomorphic to a regular subalgebra of the complex algebra of some relational structure.

Proof:

If B is a normal Boolean algebra with operators, then there exists, by the extension theorem for Boolean algebras with operators, a complete and atomic Boolean algebra with operators U which is a perfect extension of B. From definition 6.1. (iii), (iv) and (v) we can conclude that . 0 is also normal, so that the conclusion follows from lemma. 6.3.2. 0

6.4 Application of the results on Boolean algebras with operators to closure algebras and cylindric algebras

As understood from the foregoing paragraphs, a Boolean algebra with operators is a more general mathematical system than just a normal Boolean algebra (as introduced in chapter 2). To round off the ideas introduced in this chapter, we will have a very brief look at some applications of the theorems regarding Boolean algebras with operations.

Definition:

(i) Let X be a set. A map C : P(X) P(X) is a closure operator on X if, for all A,B C X ,

A C C(A), if A C B, then C(A) C C(B), C (C (A)) = C(A).

Note the intimate connection between the structure, say F(X), consisting of the closed subsets of a topological space X and the closure operator C : P(X) P(X), which maps a subset A of X to its closure, say A. Namely F(X) = {A CX I A= A} and A = E C B}.

47 (ii) An algebra U = (A, +, 0, •, 1,C) (where + and • are operations on A 2 , C is an operation on A to A, and 0 and 1 are elements of A) is called a closure algebra if it satisfies the following conditions:

(A, +, 0, • 1) is a Boolean algebra. C is an additive function on A to A. C(C(x)) = C(x) for any x E A. x C(x) = C(x) for any x E A. C(0) = 0.

The following two results, which follow easily from the above definition as well as earlier results stated in this chapter, can be seen as direct applications of the work done so far in this chapter.

Lemma 6.4.1 . For any closure algebra B there exists a complete and atomic closure algebra U which is a pe7fect extension of B.

Lemma 6.4.2 Suppose U is an arbitrary set, A is the family of all subsets of U, and R C U 2 . Then the algebra U = (A, U, 0, fl, U, R*) is a closure algebra if, and only if, R is reflexive and transitive, and the domain of R is U.

We now state and prove the following important theorem regarding closure algebras:

Theorem 6.4.3 An algebra B = (B,+,0,•,1,C) is a closure algebra if, and only if, B is isomorphic to an algebraic system B' = (B, U, 0,11, U, R*) where B is a regular set - field whose universal set is U, and R is a reflexive and transitive relation whose domain is U and which satisfies the condition: R*(X) E B for every X E B (i.e. R*!B is an operation on B to B).

Proof:

Assume that B is a closure algebra. By lemma 6.4.1, there is a complete and atomic closure algebra U which is a perfect extension of B. By the definition of the complex algebra of a relational algebra,. lemma 6.3.2, lemma 6.4.2, and the definition of a closure algebra we can conclude that U is isomorphic to an algebra U' = (A, U, 0, fl, U, R*) where U is a certain set, A is the family of all subsets of U, and R is a reflexive and transitive relation whose domain is U. It is clear that B is a regular subalgebra of LI, and hence it is isomorphic to a

48 regular subalgebra 13' = (B, U, 0, fl, U, R*) of LC. Since 13 is a Boolean algebra supplemented by a unary operation, B is a regular set - field, and R*IB is an operation on B to B.

If C3 is isomorphic to an algebra 13' with the properties listed in the theorem, then (by lemma 6.4.2) 13 is isomorphic to a subalgebra of a closure algebra and hence, by the original definition of a closure algebra, it is itself a closure algebra.

From the foregoing theorem follows the following representation theorem for closure algebras (which we just state here without any proof):

Every closure algebra B is isomorphic to an algebra 8' = (B, U, 0,11, U, C) where U is a topological space with the closure operation C, and B is a set - field whose universal set is U and which satisfies the condition: C(X) E B for every X E B.

In chapter 5 we introduced the whole concept regarding a cylindric algebra. We also proved some very important theorems which could help one represent a cylindric algebra as a less - involved mathematical system. We therefore just give here another clear and equivalent definition of a cylindric algebra and state some theorems (without any proofs) which can be seen as direct applications of the work done in this chapter (regarding Boolean algebras with operators) to cylindriC algebras.

Definition:

(i) An algebra U = (A, +, 0, ;, 1, Co , CO (where + and are operations On A 2 to A, Co and C1 are operations on A to A, and 0 and 1 are elements of A) is called a cylindric algebra of dimension 2 if the following conditions are satisfied:

1. (A, +, 0, •1) is a Boolean algebra. • 2. Co and C1 are selfconjugate functions (i.e. for any elements x, y E A, if Co (x)•y = 0, then Co (y) • x = 0, and if Ci (x) • y = 0, then Ci (y) • x = 0). Co (Co (x)) = Co(x) and Ci (C1 (x)) = C1 (x) for any x E A. Co (Ci (x)) = 1 for any element x E A with x # 0.

Lemma 6.4.4 For any cylindric algebra .8 there exists a complete and atomic cylindric algebra U which is a perfect extension of B.

Lemma 6.4.5 Suppose U is an arbitrary set, A is the family of all subsets of U, R o CU2 ,

and R1 C U2 . Then the algebra U = (A, U, 0, fl , U,11;`, RI) is a cylindric algebra if, and only if, the following conditions are satisfied:

(i) Ro and R1 are equivalence relations.

49 (ii) R0 1171 = U2 .

The next theorem, of which we omit the proof because it is very similar to that of theorem 6.4.3, can be seen as a representation theorem for cylindric algebras:

Theorem 6.4.6 An algebra B = (B,-1-,0,•,1,CO3 C1) is a cylindric algebra if, and only if, B is isomorphic to an algebraic system B' = (B, U, R,*„ RI) satisfying the following conditions:

B is a regular set - field; U is the universal set of B; R6(X) E B and RI(X) E B whenever X E B. R0 and R1 are equivalence relations. R0IR1 = U2 .

An important modification of theorem 6.4.6 is known (which, however, requires an essentially new method of arguing). It can be shown that, after omitting the term "regular" in condition (i), the following condition can be added:

R0 n R1 is the identity function on U to U.

The result thus obtained admits an interesting geometric interpretation: For, as is easily seen, it implies that the set U and the relations R0 and R1 can be chosen in theorem 6.4.6 in such a way that:

U is a set of the form U = X x Y, and hence can be regarded as an abstract two- dimensional analytic space with axes X and Y;

R0 is the relation which holds between any two couples (x, y) and (x', y') in U if, and only if, y = y', and hence Ro* coincides with the so-called operation of cylindrification parallel to the axis X; similarly,

R1 holds between (x, y) , , y') E U if, and only if, x = x', and hence R*,i coincides with cylindrification parallel to Y. In this geometric form, the modified theorem 6.4.6 motivates the choice of the name given to cylindric algebras, and it becomes the natural representation theorem for this class of algebraic systems.

Another example of a Boolea.n algebra with operators is a so-called relation algebra. This concept was defined in chapter 5. We also mentioned some important facts, regarding the representation of a relation algebra, in chapter 5. Besides this, one can (in a very similar way as we described it for closure algebras and cylindric algebras) use the work done in the first few paragraphs of chapter 6 to derive some more interesting and useful theorems regarding relation algebras. However, in this thesis we do not go into deeper detail regarding this process.

50 Chapter 7

Some applications of Representation Theorems in Universal Algebra as well as in Algebraic Logic

The whole idea of "representing" or "simplifying" an abstract mathematical structure into less-involved and "more understandable" models plays an important role throughout any mathematicians journey of studying this abstract science. Chapters 2 to 6 of this thesis studied, in detail, ways to represent various classes of Boolean algebras in order to make the stuctures "more understandable".

The representing process described. above cannot be done without the skills obtained from universal algebra incorporated in modern logic. To demonstrate these skills in the best possi- ble way, I decided to finish this thesis by describing some of the applications of representation theorems in universal algebra as well as in algebraic logic.

7.1 Applications in Universal Algebra

The whole process of building up so-called representation theorems to represent a certain mathematical structure into well-known universal structures can be seen, in its own right, as an application of the idea behind representation theorems in universal algebra. We apply representation theorems in order to be able to "conclude", "derive" or "describe" important properties of (sometimes highly demanding) universal structures. For example, one can prove or disprove that properties which hold for some groups(but not for others), hold in specific cases. This can be done by applying Cayley's theorem which states that the involved group is isomorphic to a group of permutations and then trying to prove or disprove that the specific involved properties hold in the group of permutations(which is sometimes far more easy to work with than with the original group). Other representation theorems can be applied in a much similar way. Finally, we consider the so-called Lindenbaum algebras as well as an extension of Birkhoff's representation theorem for finite distributive lattices.

51 7.1.1 The so-called Lindenbaum algebra

This paragraph does not follow directly as an application of the already existing represen- tation theorems. It rather deals with an important fragment of mathematical logic and the part Boolean algebras play in it. We do not claim to be presenting a primer on formal logic, and those unfamiliar with the subject are referred to standard texts for motivation and background.

Take an infinite set of propositional variables, denoted p,q,r,..., and define a wff (or well-formed formula) by the rules:

Any propositional variable standing alone is a tuf f ;

if w and 0 are tv f f's, so are (co A 0), (ep V v'), and (co 0);

Any tvf f arises from a finite number of appliCations of (i) and (ii).

Now we define equivalence relations and ,•-1— semantic equivalence and syntactic equivalence, on tuf f's by -d k if and only if co FE 0, if and only if FL (co 0) FL (0 —÷ So) where co a- tb means that both co 0 and 0 co are tautologies and EL (y 0) and FL (0 c9) means that co 0 and 0 co are theorems in the deduction system L of propositional calculus respectively.

Given the soundness and adequacy theorems, it is an easy exercise to show that ^- , k and

,-Fare actually the same relation. Let ti denote either or let [co] be the equivalence class of co under ti and denote the set of ,--equivalence classes by LA or, where we need to specify which relation is being used, L4 or LA F .

We show that, for either choice of there are natural operations making LA into a Boolean algebra. The most economical route is to define an order relation on LA, to show this makes LA a lattice and finally to show that this lattice is Boolean. All the verifications required are much easier for -* than for -- F . This is only to be expected. In the former case only logical equivalence and implication are involved. In the latter,' it is necessary to show that many wf f's are theorems of L. Here we give only an indication of the steps, but the complete proof also follows very easily: Define < on LAk by

[co] 5_ [0] if and only if co and on LAF by [(p] < [0] if and only if FL (y 0).

Note that, in the foregoing definition, co 0 means that, for any given assignment of values from the involved Boolean algebra to the variables occuring in co or 0, it must take the same value.

52 It can be checked in either case that < is well defined, that is, [yo] = [Sod, [0] = [O i] and [v] < [0] together imply P i] _< [01 ]. Further, < is an order relation. In (LA, there are greatest and least elements,

1 f [co] where yo is any tautology ( for [(p] where (i.o is such that FL so( for P-1-), and 0, obtained similarly, with [c,o] replaced by [---y]. The next step is to define join, meet and complement on LA. Let [co] V [0] := ko V Oh [p) A [lk] := [(to A 0] [-v]. We claim that

(A, <) is a lattice with join and meet given by V and A;

(LA, V, A) is distributive;

[y] V Fpr = 1 and [v] A [(tor = 0.

Some guidance on checking these claims for LA )- is called for. To show, for example, that V vi is the least upper hound of [yo] and [li] with respect to <, we need

EL (p —> 0)), L (0 ( -Y 0)), L (((p x) ((/, x) (( – p --+ x))) for any wf f x.

The first of these is a well-known theorem and the second is an instance of an axiom. The third is easily obtained using the deduction theorem and the following theorems of L:

FL ((a —) /3) (-1 /3 -1 a)); FL ((-ia -1/3) #) -4 a)). We conclude that each of (LAk, V, A,' , 0, 1) and (L Ah V, A,' , 0,1) is a Boolean algebra. By using the adequacy theorem, it is clear that the last two algebras are actually the same Boolean algebra. This Boolean algebra is known as the Lindenhaum algebra and has, like all other Boolean algebras, a wide spread of applications.

7.2 The Subdirect Product Representation Theorem - an extension of Birkhoff's Representation The- orem for Finite Distributive Lattices

The constructing of representation theorems (like Birkhoff's representation . theorem for finite distributive lattices - see chapter 3) to represent a certain mathematical structure (a finite

53 distributive lattice in Birkhoff's case) as a well-known mathematical model can be seen as an application of the idea behind "simplifying" structures into well-known models. However, in this paragraph we shortly describe a direct application of the skills studied. in Birkhoff's theorem for finite distributive lattices. We're namely going to state and prove a far more involved representation theorem which was established by Birkhoff in 1944 and which is based on Birkhoff's representation theorem for finite distributive lattices. Note that the definitions introduced in this paragraph are also very useful for application in earlier chapters. For further reading in connection with this theorem, "General Lattice Theory", by G. Grazer, is recommended.

Definitions:

Let pi = qi be identities for i E I and I an arbitrary set. The class K of all lattices

satisfying all identities pi = qi , i E I, is called an equational class of lattices. An equational class of algebras is defined analogously.

Let A be an arbitrary set. If Q is a binary relation on A then two elements (say a and b) are in relation with respect to Q if and only if (a, b) E Q. We denote this fact also by writing a =- b(Q).

An equivalence relation 0 on a lattice L is called a congruence relation on L iff ao b0 (0) and al b1 (0) imply that aoAai boAbi (0) and ao Va i boVb1(0). Congruences on an arbitrary algebra are defined in a very similar way.

An algebra A is called subdirectly irreducible if and only if there exist elements u, v E A such that u v and n v(0) for all congruences 0 > w, where w denotes the identity realtion.

Theorem 7.2.1 The Subdirect Product Representation Theorem Let K be an equational class of algebras. Every algebra A in K can be embedded in a direct product of subdirectly irreducible algebras in K.

Pro of:

For a, b E A, a 0 b, let X denote the set of all congruences 0 of A satisfying a 0 b(0). Since w E X, X is not empty. Let C be a chain in X. Then 0 = U(4) (1:0 E C) is a congruence, a 0 b(0), and thus every chain in X has an upper bound. By Zorn's lemma, X has a maximal element Ca , b). We claim that A / b) is subdirectly irreducible; in fact ; it = [(2]0 (a , b) and v = ' (a, b) satisfies the condition of subdirect irreducibility. Indeed, if 0 is any congruence of A I b), 0 0 w, then by the second isomorphism theorem 0 = I 11.)(a , b). Since 0 0 w, we obtain > 71)(a , b), and so a b(:I)). Thus u v(0), as claimed. Let B = fJ(AIb(a, b)la, b E A, a 0 b); then B is a direct product of subdirectly

54 irreducible algebras. We embed A into B by p : x , where h takes on the value [41 k (a , b) in the algebra Akk(a, b). Clearly, co is a homomorphism. To show that isis one - to - one, assume that fs = fy . Then x y(0(a,b)) for all a, b E A, a b. Therefore, x y(A(ka, b) I a, b E A, a b)), and so x = y. Thus the theorem holds.

We got a little bit more than claimed. If we pick x E Altk (a , b), then x = [y]7k (a, b) for some y E A. Thus there is an element in the representation of A whose image in Akb(a, b) is x; such a representation is called subdirect.

We next turn our attention to the application of so-called representation theorems in alge- braic logic specifically.

7.3 Applications in Algebraic Logic

When a person wants to solve a basic problem in Mathematics, one mostly thinks in terms of the well-known classical logical system. Throughout this thesis, we discussed the process of constructing representation theorems in order to represent abstract algebraic semantic concepts(i.e. theorems of the propositional calculus) and structures, namely Boolean alge- bras, of the classical logic system as less-involved and well-known semantic concepts and structures of the classical logic system - fields of sets in most of the cases. This proved the validity of concepts of the formal classic logic system which are defined with no reference to an interpretation, in a particular semantic context. The application of representation theo- rems as well as the role which cylindric algebras and Heyting algebras play in algebraization of the classical logic system will become more clear in the next paragraph.

From the completeness theorem follows that a sentence or a formula in the classical logic sys- tem is true or valid if and only if it is true or valid in every universal structure of the system. This means that, for instance, to conclude that a certain formula is valid in the classical logic . system, the involved sentence must be true when considering as a "property" or a "state- ment" in Heyting algebras or cylindric algebras also.Heyting as well as cylindric algebras are both very abstract universal structures and it is not always preferable to conclude the validity of formula in these structures.Nevertheless, from the completeness theorem follows that the involved sentence or formula is valid in the involved universal structure (Heyting algebras or cylindric algebras in this case) if and only if it is valid in a more describing and "less - involved" semantic system.The discussion of such systems is beyond the scope of this thesis, but readers are recommended to read the articles which were mentioned at the beginning of chapter 5. The theorems, regarding Heyting algebras an cylindric algebras(as discussed in chapter 5) play an important role when applying the completeness theorem.The first and second isomorphism theorems as well as the so - called canonical mapping between Heyting algebras and cylindric algebras also help to verify that certain properties of a statement in Heyting algebras, for example, also hold in the involved semantic system. This result in an conclusion about the validity of the original given sentence or formula.

Apart from the classic logic system, mathematicians prefer to work in other logic systems, which are different to the classic system. Intuitionistic logics is an example. A brief summary

55 of intuitionistic propositional calculus as well as the role which representation theorems play in this logic system will be discussed in the next paragraphs.

7.3.1 Intuitionism and Intuintionistic Propositional Logics

Intuitionism is one of the main philosophical trends in mathematics (the others being for- malism and logicism) which arose in the beginning of this century as attempts to overcome the crisis in the foundations of mathematics caused by paradoxes (Russell's and others) discovered in Cantor's set theory. Intuitionism goes back to Kant but as a philosophy of mathematics was developed by the Dutch mathematician L. Brauwer.

Intuitionism, which is closely related to constructivism, challenges the traditional way of dealing with and reasoning about infinite sets - especially the indirect proofs of existence by contradiction using the reduction of double negation: -,--.3x(P(x) ----4 P(x)), and the validity of the related classical law of excluded middle: P V -43. The intuitionistic ideas lead to a non-classical, constructive meaning of the logical connectives, which now can be interpreted as proof-constructions.

The first complete axiomatization of the intuitionistic logic was proposed by Arend Heyting in 1928 as a Hilbert-style sentenial calculus. Later on, in 1935, Gerhard Gentzen constructed sequential calculus and natural deduction calculus for the intuitionistic and for the classical logic.

A brief summary of Heyting's intuitionistic propositional calculus (IPC) is as follows:

Axiom schemata:

(A '—'1) --4 Y); (A —*2) ((P (0 —4 0)) —4 ((cP 0) ( 0)); (A A /) ---4 (cp A 7,b)); (AAE) yoAlk—>cp; coAlk—>0; (A V /) y V 0; --4 co V zi);

(A V E) —> 0) --÷ ((lk 0) —4 ((y 0) —4 0)); (Al) 1 —4 co;

Rule of inference:

(MP)

Definitions:

56 (i) Let F be a set of formulae and c,o a formula. An intuitionistic derivation of co from the assumptions r is a finite sequence of formulae a l , , an such that:

For every i = 1,2, ... , n, ai is either an axiom (i.e., an instance of an axiom scheme), or a i E F, or it is obtained from some aj , ak , j, k < i, by MP; an =

We denote the above definition by F 1-ipc cp and say that cp is intuitionistically deriv- able from F.

(ii) yo is a theorem of IPC if hIPC

(iii) Intuitionistic propositional logic (INT) is the set of all theorems of IPC.

(iv) An intuitionistic theory is any set of formulae T which is closed under 1-ipc.

(v) A theory T is consistent if 1 0 T . (vi) A theory T is maximal if it is consistent and cannot be extended to another consistent theory.

(vii) If F is a set of formulae, then Th(F) is the smallest theory which contains F. Th(F) is called the theory generated by F.

(viii) A set of formulae F is consistent if Th(F) is consistent.

By using basic definitions, rules and existing axioms, one can easily prove the following properties for IPC :

If (to E F then F H ipc y; If F hipc cio and F then Fipc

If F hipc (10 and F H ipc cp —i then F hipc 2,b;

If F for every E O and A 1-IPC cp then F I-Ipc cp. If (po him 7/' and V, HIPC 0 then cp 1- 113C D . (Note that this property is a special case of the foregoing one).

Note that the axiom 1 y in IPC seems perplexing and unnatural. The criticism of this and related logical laws has lead to various alternatives to the classical and intuitionistic logic. One of them is the so-called minimal logic, proposed by Johansson, which can be obtained from IPC by dropping the axiom 1 cp but keeping 1 in the language.Dropping 1 from the language, and hence too, yields the positive logic). The minimal and the intuitionistic logic are equipotent in the sense that the theorems of each of them are simply translated into theorems of the other.

57 An important application of representation theorems in this intuitionistic logic system is as follows: According to the metatheorem which states that HA cp = 1 implies Hipc tk(where HA is the variety of Heyting algebras), a formula in the INT system is a theorem if and only if it is valid in every abstract algebraic semantic structure of the INT system - Heyting algebras in this case. Nevertheless, Heyting algebras are very abstract mathematical struc- tures and mathematicians would like to interpret statements of the formal INT system in a more "discriptive" and particular semantic conctext namely relational semantics.(Note that relational semantics for propositional logics are frequently more abstract than algebraic semantics. Their advantages for the logician often result directly from their degree of abstrac- tion, which tends to correspond to a less complex proof theory). Nevertheless, the logician must be able to define the validity of a formula (of the formal INT system) interpretated in the relational semantics, i.e.: completeness must •be proved between the INT system and its particular relational semantic part. Representation theorems regarding Heyting algebras are very useful to prove this completeness. The representation theorem (of which the proof follows trivially from other representation theorems, regarding Heyting algebras, which we studied in earlier chapters) which states that every Heyting algebra can be represented as a mathematical model of the form (Ili, <, V), is of great advantage. Firstly, one can say something about the validity of a formula. (of the formal INT system) in the class of Heyting algebras by means of the existing completeness theorem. The validity of the involved formula in the class of Heyting algebras can be interpreted in the relational semantics by means of the above representation theorem. This process results in completeness regarding a formula in the formal INT system and its particular relational semantic counterpart.

Apart from intuitionistic logics there are also other logic systems which are sometimes pref .- eyed by mathematicians. The relevant logic system is an example. As in the case of the intuitionistic logics, representation theorems play a very important role in the process of building completeness between the formal system and its particular semantic counterpart.

Although one can write books on this subject, I trust that this thesis will help the reader to get more insight regarding the representation of abstract mathematical structures as well- known (and less-involved) mathematical models.

58 REFERENCES

Arnold, B.H. 1962: Logic and Boolean algebra. Englewood Cliffs, N.J.: Prentice - Hall.

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