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Representation Theorems in Universal Algebra and Algebraic Logic

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Representation Theorems in Universal Algebra and Algebraic Logic 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 .
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