STANDARD UNARY ALGEBRAS by Tracy Wall B.Sc., University ofNorthern British Columbia, 2004 PROJECT SUBMITTED IN PARTIAL FULFJLLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE INMATHEMATICS THE UNIVERSITY OF NORTHERN BRITISH COLUMBIA December 201 0 © Tracy Wall, 2010 UNIVERSITY of NORTHERN BRITISH COLUMBIA LIBRARY · Prince George, B.C. Abstract This research studies three-element unary algebras to determine which of them are standard. A quasi-variety generated by an algebra M is standard if it consists exactly of those algebraic structures being the same type as M which carry a compatible Boolean topology and are models of the quasi-equational theory of M. This work shows that two previously unclassified structures, M, are standard. 11 Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures VI 1 Introduction and Background 1.1 Introduction . 1.2 Definitions . 1.3 Background . 1 1 2 4 2 Standard Topological Structures 11 2.1 Standardness Theorems . . . . . . . . . . . . . . . . . . . . . 11 3 Example 1 3.1 P artition 1 . 3.2 Partition 2 . 3.3 Partition 3 . 3.4 Base Partition . 3.5 The Proof of Theorem 3.1 16 4 Example 2 4.1 Partition 1 . 4.2 Partition 2 . 4.3 Partition 3 . 4.4 Base Partition . 4.5 Proof of Theorem 4.1 41 Two More Examples Example 3 Example 4 . . . . 63 5 26 32 34 37 39 50 54 56 59 60 5.1 5.2 63 64 lll Summary 69 Bibliography 70 6 lV List of Tables 2.1 Three-element Unary Algebras . 14 3.1 3.2 Operations of MI . . . . . . . . Six cases for a and b, give rise to 18 situations (22 including subcases) using four partitions . 18 4.1 4.2 Operations of Ml 2 . . Four cases for a and b give rise to 9 situations (13 including subcases) using four partitions. 43 5.1 Operations of MI 4 66 . . v 21 45 List of Figures 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 .. . .. The Structure of M . . . . . The Structure of M 3 .. . . . . Clopen Sets Separating 0 and 1 with a Partition of X The Elements Z 0 and Wo . Partition 1 . Partition 2 . Partition 3 . Base Partition . 16 17 21 23 27 32 35 37 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 The Structure of M 2 The Structure of M 2 3 Clopen Sets Around 0 The Element Z Partition 1 . Partition 2 . Partition 3 . Base Partition . 41 42 46 49 50 55 57 59 5.1 5.2 5.3 5.4 The Structure of M 3 The Structure of M 3 3 The Structure of M 4 The Structure of M~ 63 64 65 65 Vl Acknowledgments I would like to thank my supervisor, Dr. Jennifer Hyndman, whose constant support, patience and guidance enabled me to develop an understanding of the subject of my project. I am grateful for the many, many hours she was willing to spend in assisting me and encouraging me towards the final completion of this work. I would also like to thank my wonderful family for all of their support through my long journey of education. V ll Chapter 1 Introduction and Background 1.1 Introduction In 2003, Clark, Davey, Haviar, Pitkethly and Talukder [3] introduced the idea of standard topological quasi-varieties. This idea was motivated by the theory of natu- ral dualities [2] which provides methods for understanding algebraic quasi-varieties, A, whenever they can be represented by a category of structured Boolean spaces. The algebraic quasi-variety A := IT§JID M, consists of all isomorphic copies of subalgebras of direct powers of M. The category A is the smallest class of algebras closed under isomorphisms, subalgebras and products containing the finite algebra M. A natural duality on A yields a category called a topological quasi-variety of the form X := IT§c JID+M where M is a finite structure such that M = (M; C, H, R, T) with operations G, partial operations H, relations R and the discrete topology T. This category X consists of all isomorphic copies of topologically closed substructures of non-zero direct powers of M. The topological structure M has the same underlying set as M and has the discrete topology. One of the benefits of duality theory is that is that the objects in X can often be much simpler than their duals in A. 1 The program of study initiated by Clark, Davey, Haviar, Pitkethly and Talukder is to determine which topological quasi-varieties are standard and which are not. By defining the quasi-equational theory of X, that is, the set of all quasi-equations satisfied by M, a nice axiomatic description of the members of X is obtained. The quasi-equational theory is denoted Thqe (M), and ModT Thqe (M) is the class of all topological models of the quasi-equational theory of M. The category X is a standard topological quasi-variety provided it consists exactly of those algebraic structures being the same type as M which carry a compatible Boolean topology and are models of the quasi-equational theory of M. The purpose of this research is to determine which topological three-element unary algebras are standard. I was able to show that M = ({0, 1, 2}; F; T) where F = {002 , 110} , and M 2 = ({0, 1, 2}; F; T) where F = {001, 002}, are standard. See Theorem 3.1 in Chapter 3 and Theorem 4.1 in Chapter 4 for these results. 1.2 Definitions The following definitions are taken fr om [1] An algebra is a set A, together with a collection of operations on A. An n-ary operation (or function) on A is any function f from An to A where n is the arity (or rank) of f. A finitary operation is an n-ary operation, for some n. The image of (a1 ... an) under an n-ary operation f is denoted by f(a 1 ... an)· An operation f on A is called a nullary operation (or constant) if its arity is zero . An operation f on A is unary, binary, or ternary if its arity is 1, 2, or 3, respectively. An algebra A is unary if all of its operations are unary, and it is mono-unary, or a unar, if it has just one unary operation. A language (or type) of algebras is a set F of function symbols such that a 2 nonnegative integer n is assigned to each member f of F . If F is a language of algebras then an algebra A of type F is an ordered pair (A, F) where A is a nonempty set and F is a family of finitary operations on A indexed by the language F , such that corresponding to each n-ary function symbol f in F there is an n-ary operation fA on A. The set A is called the universe or underlying set of A = (A; F) and the fA's are called the fundam ental operations of A. For example, a monoid is an algebra (M, ·, 1) with a binary and a nullary operation satisfying x · (y · z) ~ (x · y) · z, and x · 1 ~ 1 · x ~ x for each x, y, z E M . Let A and B be two algebras of the same type. Then B is a subalgebra of A if B ~ A and every fundamental operation of B is the restriction of the corresponding operation of A, that is , for each function symbol f, f 8 is fA restricted to B; we write simply B ~ A. Where the context is clear, we drop the superscript in fA and write f. A subuniverse of A is a subset B of A which is closed under the fund amental operations of A , that is, if f is a fundamental n-ary operation of A and a 1 ... an E B, we would require f(a 1 ... an) E B. Thus if B is a subalgebra of A; then B is a subuniverse of A. An algebra A is locally finite if, given any set C ~ A, the subalgebra generated by C is finite. A class K of algebras is locally finite if every member of K is locally finite. An example of a locally finite algebra is any algebra wit h a finite underlying set and finit ely many operations. Let A and B be two algebras of the same type F. A mapping a : A ---t B is a homomorphism from A ---t B if it preserves the operations in F, that is for every n-ary f E F and for a 1 , ... , an E A we have ajA(a1, ... , an)= f 8 (aa1, . .. , aan) Such a mapping a: A ---t B is an isomorphism, that is A is isomorphic to B, if a is 3 one-to-one and onto. A variety is a non-empty collection of algebras of a fixed type that is closed under homomorphic images, subalgebras and products. A quasi-variety is a nonempty collection of algebras of a fixed type that is closed under isomorphic images, subalgebras and products. A quasi-atomic formula is an expression which is either an atomic formula, a neg-atomic formula, •a, or an implication, {31 1\ ... 1\ f3m ====? a, where m ~ 1 and {3 1 , ... , f3m, a are atomic formulae . A quasi- equation is a universally quantified formula of the form ¢YI 1\ (h 1\ ... I\ ¢m ====? a, where the cPi are atomic equations. For an algebraic language, the cPi are simply equations. The quasi-atomic theory of M is denoted by Thqa(M). For unary algebras, where there are no relations, then Thqa(M) is called the quasi-equational theory of M and is denoted Thqe (M). The quasi-equational theory of a class of a topological structures M is the set of quasi-equations that hold and are satisfied by M, and M is a model of a set of quasi-equations if it satisfi es every quasi-equation in the set. The quasi-variety A generated by an algebra M is simultaneously the class of all algebras of the same type as M that are models of the quasi-equational theory of M and is the smallest class of algebras containing M that is closed under isomorphic copies, subalgebras and products. This quasi-variety is denotedll§JPl(M). Note t hat if M is finit e with a finit e language, then ll§JPl(M) is locally finite. 1.3 Background An area of on-going research in the theory of natural dualities is the problem of exactly which finite algebras generate a dualizable quasi-variety. Clark and Davey [2] have shown that two-element algebras are clualizable. The dualizability problem 4 for three-element algebras, however, was more complicated but was later solved by Clark, Davey and Pitkethly [4]. Their classification system divid es three-element unary algebras into zero, one, two and three-kernel algebras. This classification is important for the dualizability of three-element unary algebras and is useful in determining which are candidates for standardness. The definition of kernel from [4], is as follows: A kernel of M is an equivalence relation on M of the form ker(u), for some unary term function u of M that is neither a constant map nor a permutation. Where the triple abc denotes the map u from M to M with u(O) = a, u(1) = b and u(2) = c, the kernel of 110 is {{0, 1}, {2} }. An algebra M = (M; F) is nkernel if there are n distinct kernels for the non-constant, non-permutation opera- tions in F. For example, a 0-kernel algebra has operations that are permutations or constants, ( {0, 1, 2}; 210, 111), and a 1-kernel algebra has operations that are permutations or constants, or operations with one fixed non-trivial kernel, e.g., ({0, 1,2};111,000,110,002). Clark, Davey and Pitkethly [4] classified the dualizable three-element unary algebras as follows: Theorem 1.1. Let M be a three-element unary algebra, on the set {0, 1, 2} 1. If M is a zero -kernel or a one-kernel algebra, M is dualizable; 2. If M is a two-kernel algebra with kernels 0112 and 02j1 , then M is dualizable if and only if: (a) pp1 and pqp are term functions of M with p, q E M and p =/: q, but 010 or 002 is not a term function of M; (b) 010, 001 and 110 are term functions of M, but 222 isn't; 5 (c) 002,020 and 202 are term functions of M but 111 isn't . 3. If M is a three-kernel algebra, then M is not dualizable. It is this type of theorem that I am looking for in classifying which algebras are standard. Given an algebraic quasi-variety A := ll§JID M, the Algebraic S eparation Theorem shows that the members of A are characterized by the existence of separating homomorphisms into M. Theorem 1.2 (Algebraic Separation Theorem). [2] An algebra A is in A := ll§JIDM if and only if, for each a, bE A where a=/: b, there is a homomorphism Uab : ~ M such that Uab (a) =/: Uab (b) . The ll§JID Th eorem shows that every algebraic structure in A is a model of the quasi-equational theory of M. Theorem 1.3 (ll§JID-Theorem) . [2] Let M be a finite algebra. For any algebra B of the same type as M, the follo wing are equivalent: (i) B E A := ll§JID M; (ii) B is a model of the quasi- equational theory of M; (iii) B is obtainable from M by repeated applications of ll, § and JID. In particular, A := ll§JID M is exactly the quasi-variety generated by M. The spaces I am working with are struct ured Boolean spaces, defined as follows, beginning with the definition of a Boolean space [2] . A topological space X is a set X, together with a collection, T, of open subsets of X, such that T includes 0 and X and is closed under finite intersections and 6 arbitrary unions. The collection T is called the topology on X. A subset of X is closed if its complement is open, and it is clopen if it is both closed and open. A collection C of open subsets is a cover of X if the union of the elements of C is equal to X. A topological space is compact if every open cover contains a finite subcover. A topological space is Hausdorff if for all a, bE X, there exist open sets U, VEX with a E U, b E V, such that U n V = 0. If U and V can always be chosen to be clopen, then X is said to be totally disconnected. A Boolean Space is a topological space that is compact, Hausdorff and totally disconnected. A structured topological space is defined in [2] to be a structure of the type (G,H,R), such that where (i) gx consists of an n-ary total operation gx xn ---? X for each n-ary total operation symbol g E G, (ii) hx consists of an n-ary partial operation hx : dom(hx) ---? X for each n-ary partial operation symbol h E H, where dom(hx) ~ xn, (iii) rx consists of an n-ary relation rx ~ xn on X for each n-ary relation symbol r E R, (iv) (X, Tx) is a topological space. The superscripts are omitted when the context is clear. A Boolean structure [3], is a topological structure X = (X; e x, Hx, Rx, Tx), such that (i) (X; Tx) is a Boolean space, 7 (ii) if h E G U H is n-ary, then the domain dom(h") is a closed subset of xn and h" : dom(hx) -----. X is continuous, and (iii) if r E R is n-ary, then rx is a closed subset of xn. Given a structured topological space, M = (M; G, H, R , T), where M is finite and the topology T is discrete, we generate the class X := IT§c JP>+M. Because M is finite and the topology is discrete, M is Boolean, so the topology on every member of X is Boolean and therefore X := IT§c JP>+M, is a category of structured Boolean spaces.[2] The following definitions explain a morphism between structured topological spaces and a substructure of a topological space. Given a structured topological space X = (X; ex, H x, R x, Tx), and another structured topological space Y = (Y; G", H" , R", T "), a continuous map c.p : X -----> Y is a morphism if (i) for each n-ary g E G and each (x1, x 2, ... , Xn) E X, we have (ii) for each n-ary h E H and each (x 1, x 2, ... , xn) E dom(hx) we have (c.p(x1), c.p(x2), ... , c.p(xn)) E dom(h") and c.p(hx(x1, X2, ... , Xn)) = h"(c.p(xl), c.p(x2), ... , c.p(xn)), (iii) for each n-ary r E Rand each (x 1, x 2, . . . , Xn) E rx we have (c.p(x1), c.p(x2), ... , c.p(xn)) E r", and (iv) c.p is continuous. 8 The structure Y = (Y; Q'lf, H'lf, R'if, T'if) is called a substructure of the structured topological space X = (X; ex, HX, Rx, Tx), written y :::; X, provided that y ~ X and (i) for each n-ary g E G the operations g'lf and gx agree on yn, (ii) for each n-ary hE H, we have dom(h'lf) = dom(hx) n yn, and h'lf agrees with hx on this set, (iii) for each n-ary r E R, we have r'lf = rx n yn, and (iv) T'lf is the relative topology obtained from Tx. A topological structure M is then said to be standard if every Boolean model of the quasi-equational theory of M is isomorphic to a closed substructure of a non-empty product of M, that is, For topological structures, in order to ensure the correct description of the members of X , Theorem 1.2 and Theorem 1.3 are modified. The Topological Separation Theorem characterizes the existence of separation morphisms into M and shows that any structure meeting a particular description is in X. The Preservation Theorem shows that every topological structure in X is a model of the quasi-equational theory ofM. Theorem 1.4 (Topological Separation Theorem). [3] Let M = (M; G, H, R, r) be a finite structure, let X := IT§c JPl+M, and let X be a compact topological structure of the same type as M. Then X E X if and only if there is at least one morphism from X to M and the following conditions hold: (i) for each x, y EX where x-=/=- y, there is an a: X--+ M such that a(x) -=/=- a(y), 9 (ii) for each n-ary h E H and (x 1 , x 2 , ... , Xn) E Xn\ dom (hx), there is an a ~ M such that (a(x1), a(x2), ... , a(xn)) ~ dom (hM), (iii) for each n-ary r E R and (x 1 , x 2 , ... , Xn) E Xn\rx, there is an a : X ~ M such that (a(xi), a(x 2), ... , a(xn)) ~ rM. Theorem 1.5 (Preservation Theorem). [3] Let M = (M; G, H, R, T) be a finite structure and let X := ll§c JID+M. Then every member of X is a Boolean model of the quasi-equational theory of M, in symbols, X ~ Modr Thqe (M). Since unary algebras have only unary operations, that is , there are no partial operations or relations, Theorem 1.4 and Theorem 1.5 are modified so that M = (M; G, T) and in Theorem 1.4, (i) holds and (ii) and (iii) do not apply. Theorem 1.6 (Separation Theorem for Unary Algebras). Let M = (M; G, T) be a finite structure, let X := ll§c JID+M, and let X be a compact topological structure of the same type as M. Then X E ...-1' if and only if there is at least one morphism from X toM, and for each x, y EX where x =/:. y, there is an a: ~ M such that a(x) =/:. a(y) Theorem 1. 7 (Preservation Theorem for Unary Algebras). Let M = (M; G, T) be a finite structure and let X := ll§c JID+M. Then every member of X is a Boolean model of the quasi-equational theory of M, in symbols, X ~ Mod 7 Thqe (M). 10 ~ Chapter 2 Standard Topological Structures 2.1 Standardness Theorems This research studies three-element unary topological algebras to determine which of them are standard. See Table 1 for a list of the three-element unary algebras. The general method is as follows: Given a, b, E X := TI§c JP>+M, the topological quasi-variety, if X can be partitioned into clopen sets with a and b in different sets and the clopen sets manipulated so that they behave like elements of an algebra in the quasi-variety, t hat is, t he clopen sets are isomorphic to a finite subalgebra of a power of M, then the algebra is standard. The following material shows that this algorithm is valid. The following Lemma from [3] says that to show X is standard it is sufficient to show for any X E X that there is a finite substructure, Y E TI§c JP>+M that satisfies the quasi-equat ional theory of M with the required separating morphism from X --+ Y. Lemma 2.1. [3] Let M be a finite structure. Then X := TI§c JP>+M is standard provided that, for every X E Modr Thqe (M), (i) for each x, y E X where x :/: y, there is a finite Y E X and an a: X--+ Y such 11 ~ that a(x) =I a(y), (ii) for each n-ary h E H and (xl, X2, ... 'Xn) E xn\ dom (hx.), there is a finite Y E X and an a: X --t Y such that (a(x 1 ), a(x 2 ), ... , a(xn)) ~ dom (h 'If), (iii) for each n-ary r E R and (xl, X2, . .. 'Xn) E xnvx, there is a finite y E X and an a: X --t Y such that (a(x 1 ), a(x 2 ), ... , a(xn)) ~ r'!f . Again, since the applications of this research are for total algebras, M! = (M; G, T), with no partial operations or relations, items (ii) and (iii) of Lemma 2.1 are not neeessary: Corollary 2.2. [3] Let M be a finite total structure. Then X := IT§c JP>+M is standard provided that, for every X E Modr Thqe (M!), and for each x, y E X where x =I y, there is a finite Y E X and an a: X --t Y such that a( x) =I a(y). Before stating the standardness theorem for unary algebras, we require the following lemma. Lemma 2 .3 . Let X be a Boolean model of the quasi-equational theory of M! and assume Y is a partition of X into finitely many clopen sets where Ux denotes the clopen set in Y containing x. Let f E F. If for all x E X we have j'X.(Ux) ~ Uf x(x)> then /'': Y --t Y defined by/"' (Ux) = U1x(x) is a well defined map. Proof. Assume Ux = Uy for some x, y EX. We have x, y E Ux so f x.(y) E fx.(Ux) ~ UJ"·(x)· Thus Ufx(y) = Utx(x)> that is, f'(Uy) = f'!f(Ux)· Therefore f': Y --t Y is well defined. 0 The algorithm required is then summarized by the following theorem. Theorem 2 .4 (Standardness Theorem for Unary Algebras) . If for every Boolean model X of the quasi-equational theory of M!, and every pair of elements a, b E X with a =I b, there is a partition Y of X into finitely many clopen sets, such that 12 (i) for all x E X the clopen set containing x is Ux, (ii) Ua nUb = 0, (iii) fx(Ux) ~ UJ'f·(x) for all X E X, (iv) f'il(Ux) := Ufx(x) for all x EM and f E F, and (v). (Y, F) satisfies the quasi-equations of M, then M is standard. Proof. Let Y be the finite partition of clopen sets chosen above with Ua 1:. Ub · The map a: X---> Y is defined by a(x) is the clopen set Ux containing x. For all x EX the map a is well defined since for each x, there is one clopen set containing x. The map CY is operation preserving because for all f E F, J(a(x)) = f(Ux) = uf(x) = a(J(x)). Therefore, CY is a morphism. The map CY is continuous because Y is finite and has the discrete topology and a- 1 ( {Uy}) = Uy, a clopen set . By Lemma 2.3 each f' is well defined. By Theorem 1.2, there is a separating morphism f3: (Y, F) ----. (M, F) such that f3(x) 1:. f3(y). Since 1{ is a finite structure and (Y, F) satisfies the quasi-equations of M, then by Theorem 1.3, (Y, F) E IT§lfD( (M, F)), but the finite elements ofll§lfD( (M, F)) are in IT§c IfD+M when endowed with the discrete topology. As f3 is continuous when Y and M have the discrete topology, by Theorem 1.6, Y E IT§c IfD+M. Therefore (Y, F, T) E IT§c IfD+M, and by Corollary 2.2, M is standard. 0 Some results are known and are shown in Table 2.1. For example, Clark, Davey, Haviar, Pitkethly and Talukder [3] showed that finite mono-unary topological algebras are standard. Also, Hyndman and Pitkethly [5] have shown two algebras are 13 standard using compactness and one is standard and injective in the topological equational class, and three algebras are non-standard using an inverse limit technique. Chapters 3 and 4 provide proofs that two additional algebras are standard. The three-element unary algebras, up to isomorphism, are listed in Table 2.1. That this list is complete must be confirmed. 1 generated monoids 2 generated mono ids 3 generated monoids 4 generated monoicls Generators 002 112 001 110 220 221 002, 112 001, 002 110, 112 001,110 001 , 112 Kernels 1 1 1 1 1 1 1 1 1 1 1 Elements 110, 002 002, 221 1 1 000, 111 112, 220 001 , 220 1 001, 221 1 010, 011 010 , 110 101 , 220 2 2 2 000, 111 , 222, 110, 002 000, 111, 222, 110, 112 001, 002, 111 001, 110, 112 001, 002 , 112 002, 110 , 112 001, 002, 22 1 001, 002, 110, 112 2 2 1 2 1 1 000 111 002 112 000 111 000, 111 000, 111 000, 111 000, 111, 222, 010 , 002 000 , 111 000, 111 000 , 111 000 , 111 ALL 000, 111 Knowledge Standard- mono-unary Standard- mono-unary Standard- mono-unary Standard- mono-unary Standard- mono-unary Standard- mono-unary Standard- needs compactness Standard - injective in the topological equational class Standard - currently uses compactness V non-standard - inverse limits L non-standard - inverse limits D non-standard - inverse limits Table 2. 1: Three-element Unary Algebras 14 The proofs in Chapters 3 and 4 frequently require the following lemmas which follow directly from the topology of Hausdorff spaces. Lemma 2.5. Let X be a Boolean model of the quasi-equational theory of M. Given a, bE X, and a=/: b, there exist clopen sets U, V with a E U, bE V and U n V = 0. In addition, given {c, d} disjoint from {a, b}, we may assume c, d tJ. U and c, d tJ. V. Lemma 2.6. Given A <:;:; X closed and b E X such that b tJ. A, there exist clopen sets U, V with U n V = 0, A <:;:; U and b E V. Lemma 2.7. Given A and B closed and disjoint with B <:;:; Ua and Ua clopen, there exists a clopen set U with B <:;:; U <:;:; U0 and UnA= 0. The next two chapters provide two examples of topological structures that are standard. These are new results. 15 Chapter 3 Example 1 The first example we consider is the structure M = ({0, 1, 2}; F; T) where F = {002, 110} and T is the discrete topology. When g denotes the operation 002 and p denotes the operation 110, the structure M is shown in Figure 3.1. @ ..._____ 9 ___ 0 Figure 3.1: The Structure of M Applying p and g to the 27 elements of M 3 yields Figure 3.2. Knowing the structure of M 3 assists in understanding three variable quasi-equations. The first result on standardness of unary algebras is the next theorem. Theorem 3 .1. The structure M = (0, 1, 2; F; T) where F = {002, 110} and T zs the discrete topology, is standard. The proof of Theorem 3. 1 uses Theorem 2.4. To show that a particular structure M is standard, we construct a finite cover of clopen sets of each Boolean model 16 (20r:jl 112 212 ~ )p p( ~ 221 121 001:) 211 Figure 3.2: The Structure of M 3 X of the quasi-equational theory of M in which particular points are separated, and the clopen sets satisfy the quasi-equational theory of M. Choose X such that X E Modr Thqe (M), that is, X is a Boolean model of the quasi-equational theory of M, and then show that X E ll§c JP+M. There are a number of cases to consider. Theorem 3.4 provides the basis for the choice of cases. Note that M has two constant valued fuctions, 0 and 1. The equations and quasi-equations satisfied by M include: (3.1) gn(x) ~ g(x), n 2: 1 (3.2) gnp(x) ~ 0, n 2: 1 (3.3) 17 gpn(x) ~ 0, n 2: 1 (3.4) pg(x) ~ p(x) (3 .5) png(x) ~ 1, n > 1 (3.6) p(x) ~ 1 <===? g(x) ~ 0 (3.7) p(x) ~ 0 => g(x) ~ x (3.8) g(x) ~ g(y) <===? p(x) ~ p(y) (3.9) p(x) ~ (3.10) ~ 1 That these quasi-equations hold can be seen from Table 3.1. p g p2 g2 pg gp pg2 p"l.g gp2 pgp gpg 1 0 1 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 2 0 2 1 0 1 2 0 0 0 0 X Table 3.1: Operations of M Throughout the remainder of Section 3.1, X is a Boolean model of the quasiequational theory of M. The following lemmas provide useful properties of subsets of X. Lemma 3.2. Assume A, B <;:;;X. (i) If A <;:;; p- 1 (B), then g(A) <;:;; p- 1 (B). (ii) If A <;:;; g- 1 (B), then g(A) <;:;; g- 1 (B). (iii) If A <;:;; p- 1 (B) and 1 E B, then p(A) <;:;; p- 1 (B). (iv) If A <;:;; g- 1 (B) and 0 E B, then p(A) <;:;; g- 1 (B). 18 Proof. For property (i) assume A ~ p- 1 (B), so p(A) ~ B. By Quasi-equation 3.5, p(A) = pg(A). This implies pg(A) ~ B and g(A) ~ p- 1 (B). Similarly, for property (ii), if A ~ g- 1 (B) then g(A) ~ B. Then Quasi-equation 3.2 implies g2 (A) ~ B, so we have g(A) ~ g- 1 (B). For property (iii), assume A ~ p- 1 (B). If 1 E B, then Quasi-equation 3.1 gives pn(x) ~ 1, which implies X= p- 2 (B) and p(X) ~ p- 1 (B). Therefore, p(A) ~ p- 1 (B). Finally, for property (iv), if A ~ g- 1 (B) and 0 E B, then Quasi-equation 3.4 gives gp(x) ~ 0, which implies X= p- 1 g- 1 (0) ~ p - 1 g- 1 (B). Thus A ~ p- 1 g- 1 (B), and we have p(A) ~ g- 1 (B). 0 Lemma 3.3. Assume A, B ~ X and An B = 0. Then p- 1 (A) n p- 1 (B) = 0. Proof. Let d E p - 1 (A) n p- 1 (B). Then p(d) E A and p(d) E B, but An B = 0, a contradiction. Therefore p- 1 (A) n p - 1 (B) = 0. 0 For each a E X, there are six possibilities and these are described in the next Theorem. Theorem 3.4. For each a E X, a Boolean model of the quasi-equational theory of MI, one of the following cases holds. (i) a= 1; or (ii) a= Oi or (iii) a tf. {0, 1}, p(a) = 1, g(a) =I= ai or (iv) {a,p(a)} disjoint from {0, 1}, g(a) = ai or (v) {a,p(a)} disjoint from {0, 1}, g(a) =I= ai or (vi) a tf. {0, 1}, p(a) = 0, g(a) =a. Moreover, p( a) =/= 1 when a =/= 1. 19 Proof. If a= 1, then Case (i) holds. If a= 0, then Case (ii) holds. Now assume a =f. 0 and a =f. 1. It is sufficient to consider the cases when p(a) = 1 or p(a) =f. 1. If p(a) = 1, then Quasi-equation 3.7 gives g(a) = 0, which is Case (iii) . If p(a) = 0, then from Quasi-equation 3.8 we know that g(a) = a, which is Case (vi). Now assume p(a) =f. 0 and p(a) =f. 1. It is sufficient to consider the cases when g(a) =a or g(a) =f. a . The former condition gives Case (iv) and the latter gives Case (v). Finally, if p(a) = 1, Quasi-Equation 3.10 gives a= 1. 0 Based on the six cases in Theorem 3.4, for distinct a and b, there are potentially 36 cases. By symmetry, we need consider 21 cases, which further reduces to 18 cases that can be described by one of four partitions. The possible partitions are: Partition 1 : X= Z U W U AU B (J K (J p - 1 (A) (J p - 1 (B) (J p- 1 (K) (J p- 1 (Z). Partition 2 : X = Z UW Partition 3 : X= Z U W U L U CUD U R (J p - 1 (Z) Base Par-t·i tion 4: X = Z (J W (J L (J p- 1 (L) (J p- 1 (Z). u L up - (Z) u R u (p- (L)\R) 1 1 Each of the 18 cases is covered by one of the four partitions. This is shown in Table 3.2. The following lemmas show the construction of clopen sets that separate the constant valued functions 0 and 1. The illustration of these sets is shown in Figure 3.3. The partition illustrated in Figure 3.3 underlies all other partitions constructed. 20 a 1 1 2 3 4 5 6 - b 2 3 4 4 - 4 1 - - 4 4 4 4 1,2 - 5 4 4 1,4 1 1,3 6 4 4 1,4 4 4 - - Table 3.2: Six cases for a and b, give rise to 18 sit uations (22 including subcases) using four partit ions. Figure 3.3: Clopen Sets Separating 0 and 1 with a Partition of X Lemma 3 .5 . Ther-e ar-e clopen sets U and V with 0 E U, 1 E V and U n V = 0. In addition, given a rf. {0, 1} and b rf. {0, 1}, we may assume {a, b} rf. U and {a, b} rf. V . Mor-eover, the clopen sets Zo and W 0 , defined as Zo = U np- 1 (V) n g- 1 (U) np- 1 (X\U) n g- 1 (X\ V) Wo = V n p- 1 (V) n g- 1 (U) n p- 1 (X\U) n g- 1 (X\ V). satisfy the following containments: 21 2. Zan Wa = 0; 3. g(Za) ~ Zo; 4. g(Wa) ~ Zo; 5. p(Za) ~ Wo; 6. p(Wo) ~ Wa· Moreover, Za = Zan p- 1 (Wa) n g- 1 (Za) n p- 1 (X\Zo) n g- 1 (X\Wo) Wa = Wa np- 1 (Wa) n g- 1 (Zo) np- 1 (X\Zo) n g- 1 (X\Wo) Proof. The existence of U and V follow from Lemma 2.5. Since Za is contained in each of p- 1 (V), g- 1 (U), p- 1 (X\U) and g- 1 (X\ V), Lemma 3.2 gives g(Za) ~ p- 1 (V), g- 1 (U), p- 1 (X\U) and g- 1 (X\V). Similarly, since Wa is contained in each of p- 1 (V), g- 1 (U), p- 1 (X\U) and g- 1 (X\ V), Lemma 3.2 gives g(Wa) ~ p- 1 (V), g- 1 (U), p- 1 (X\U) and g- 1 (X\ V). Thus we have g(Za) ~ Za and g(Wa) ~ Z 0 , as Z 0 ~ g- 1 (U) and W 0 ~ g- 1 (U). By Lemma 3.2, p(Wa) is contained in each of p- 1 (V), g- 1 (U) , p- 1 (X\U), and g- 1 (X\ V), and p(Za) is contained in each of p- 1 (V), g- 1 (U), p- 1 (X\U), and g- 1 (X\ V). Thus we have p(Wa) ~ W 0 and p(Za) ~ Wa as Wa ~ p- 1 (V) and Z 0 ~ p- 1 (V). The last two equalities of the Lemma hold because Wa ~ V ~ X\U ~ X\Zo and Za ~ U ~ X\ V ~ X\ W 0 . 0 22 p Figure 3.4: The Elements Z 0 and Wo The behaviour of g and p on Z 0 and W0 is illustrated in Figure 3.4. If X = Z 0 U W 0 , then X has been partitioned by clopen sets as required by T heorem 2.4. Now assume Q = X\(Zo U Wa) is non-empty. We need to construct 1 more clopen sets that cover X\(Zo U W 0 ) . Consider g- 1 (Z 0 ), g- 1 (W0 ), p- (Zo) and Lemma 3.6 . There exists W a clopen subset of W 0 , and Z a clopen subset of Zo such that p(p- 1 (Wo)\g - 1 (Zo)) n W = 0; g(g- 1 (Zo)\p - 1 (Wo)) n Z = 0; p(p- 1 (Wo) n g- 1 (Zo)) <;;;;; W; g(g - 1 (Zo) n p - 1 (Wa)) <;;;;; Z. Moreover·, the sets Z and W satisfy the following containments: 1. 0 E Z and 1 E W; 2. z n w = 0; 3. g(Z) <;;;;; Z; 4. g(W) <;;;;; Z; 5. p(Z) <;;;;; W; 6. p(W) <;;;;; W; 23 Proof. The existence of W and Z satisfying the displayed equations, is guaranteed by Lemma 2.7. Given 1 E p(p- 1 (Wa)\g- 1 (Za)), this implies there exists an a E p- 1 (Wa)\g- 1 (Za) with p(a) = 1. With g(O) = 0 and 0 E Z 0 and with p(O) = 1 and 1 E W 0 , we have 0 E g- 1 (Za)np- 1 (W0 ). Therefore 1 = p(O) E p(g- 1 (Za) E g(g- 1 (Xo) n p- 1 (Wa)) ~ 0 = g(O) n p- 1 (Wa)) ~ Z. It then follows with Zan W = 0, that Z n W = 0. 0 We have Z ~ Za ~ g- 1 (Za) n p- 1 (Wa), and this implies g(Z) ~ g(Za) ~ g(g- 1 (Za) n p- 1 (Wa)) ~ Z. Therefore , g(Z) ~ Z. Similarly, Z ~ p- 1 (W0 ), so p(Z) ~ W. With W ~ Wa ~ g- 1 (Za) n p- 1 (Wa), we have g(W) ~ g(Wa) ~ g(g - 1 (Za) n p- 1 (Wa)) ~ Z . Therefore g(W) ~ Z. Similarly, we have p(W) ~ W. In order to prove the remaining containments, we first show that g- 1 (Z) g- 1 (Za) n p- 1 (Wo) = p- 1 (W). By choice of Z, we have g- 1 (Za) n p- 1 (Wo) ~ g- 1 (Z). To show g- 1 (Z) = g- 1 (Zo) n p- 1 (vV0 ), consider that a E g- 1 (Z). Then g(a) E Z ~ Zo and a E g- 1 (Z) ~ g- 1 (Z0 ). We need to show a E p- 1 (W0 ). If a F/:. p- 1 (W0 ), then a E g- 1 (Za)\p- 1 (Wa) and g(a) E g(g - 1 (Za)\p - 1 (Wa)). But g(g - 1 (Za)\p - 1 (Wa)) n Z = 0, a contradiction. Therefore a E p-1(W0 ). So we have g- 1 (Z) ~ g- 1 (Za) n p- 1 (W0 ) . Similarly, claim p- 1 (W) = g- 1 (Za) n p- 1 (W0 ). Certainly p- 1 (W) ~ p- 1 (W0 ). Then let a E p- 1 (W). We need to show a E g - 1 (Z0 ). If a ¢:. g- 1 (Z0 ), then a E P- (Wa) \g- 1 (Za), which implies p( a) E p(p- 1 (Wo) \g- 1 (Za)) . But p(p- 1 (Wo) \g- 1 ( Z 0 )) 1 n W = 0. So we have a E g- 1 (Z 0 ). 0 Lemma 3.7 . For Z and W with the properties described in Lemma 3.6, the sets satisfy: 24 (i) g- 1 (W) = 0; (ii) p- 1 (W) =/= 0 and g- 1 (Z) =/= 0. 1 Proof. Suppose g- 1 (W) =/= 0, then there is some element c E g- (W), that is, g(c) E W. Then by Quasi-equation 3.2, g(c) = g 2 (c) E g(W). By Lemma 3.5, g(W) ~ Z, so g(c) E g(W) ~ Z. Therefore, g(c) E Z which implies that g(c) E W nz, a contradiction since W n Z = 0. Therefore, g- 1 (W) = 0. Now by Lemma 3.5, we have 1 E Wand 0 E Z. It follows from Quasi-equation 3.1, that p(O) ~ 1 and 1 E W. So 0 E p- 1 (1) ~ p- 1 (W). Similarly, g(1) ~ 0 and 1 0 E Z. Then g- 1 (1) ~ g- 1(Z) and we have 1 E g- 1(Z). Therefore p- (W) and g- 1 (Z) are non-empty. 0 1 An additional set that is part of the clopen cover of X is L = p- (W)\(Z U W). Lemma 3.8. The sets L, Z and W satisfy (i) p(L) ~ W and g(L) ~ Z; (ii) p(p- 1(L)) ~ L and g(p- 1(L)) ~ p- 1(L); (iii) g- 1 (L) = 0. Proof. We have L ~ p- 1(W) = g- 1 (Z), so L ~ p- 1(W) and L ~ g- 1 (Z). Then it follows that p(L) ~ W and g(L) ~ Z. As p- 1(L) ~ p- 1(L), by Lemma 3.2, g(p- 1(L)) ~ P-1(L). 1 Now, assume e E g- 1(L). Then g(e) ELand we have g(e) E p- (W) and g(e) ~ (ZUW). But p- 1 (W) = g- 1 (Z), so g(e) E g- 1 (Z) and g 2 (e) E Z. Quasi-equation 3.2 implies g(e) E Z, a contradiction since g(e) tf:. (Z U W). Therefore, g- (L) = 0. 1 1 Lemma 3.9. Z satisfies, p(p- 1 (Z)) ~ Z and g(p- 1 (Z)) ~ p- (Z). In addition, 25 0 (i) p- 1 (Z) n (Z u W) = 0; (ii) p - 1 (Z) nL = 0. Proof. If p- 1 (Z) = 0, then the claims are vacuously true, so assume p - (Z) -/= 0. 1 As p- 1 (Z) ~ p- 1 (Z), it follows that p(p- 1 (Z)) ~ Z, and by part (i) of Lemma 3.2, g(p- 1 (Z)) ~ p - 1 (Z). By Quasi-equation 3.5, pg(p- 1 (Z)) = p(p- 1 (Z)) ~ Z. Since p(p- 1 (Z)) ~ Z and we know p(Z) ~ Wand p(W) ~ W, then p(Z U W) ~ W. But this implies p(p- 1 (Z) n (Z U W)) ~ p(p- 1 (Z)) n p(Z U W) ~ Z nW, but Z n W = 0. Therefore p- 1 (Z) n (Z U W) = 0 as claimed. Now L = p- 1 (W) \(Z U W). With Z n W = 0, by Lemma 3.3, we get p - (Z) n 1 p - 1 (W) = 0. 0 Accordingly, there are disjoint clopen sets around the elements 0 and 1 in which particular points are separated from 0 and 1, such that and t he clopen sets satisfy the quasi-equational theory of .MI. The next four sections illustrate the construction of the four partitions of X. 3.1 Partition 1 In this section the first non-trivial partition of X into clopen sets is constructed. This partition is larger than the base partition and is illustrat ed in Figure 3.5. Theorem 3.10. Assume a,b EX with {a,b} disjoint from {0, 1} and a-/= b, and a, b satisfy one of 1, 2, or 3, below. Th en there exist disjoint clopen sets, A, B, Z, W and L, such that Z, W and L satisfy the properties of Lemma 3.6 to Lemma 3.9, and X can be partitioned as 26 F igure 3.5: Partition 1 where the partition separates a, b, 0 and 1, and the sets satisfy the quasi-equations ofM. 1. p(a) = 1 and g(a) =fa, with either (a) p(b) = 1 and g(b) =I b; or (b) p(b) rt {0, 1, a}, g(b) = b; or (c) p(b) tJ_ {0, 1, a}, g(b) =f b. 2. p(a) tJ_ {0, 1}, g(a) =a and p(a) =f p(b), with either (a) p(b) tJ_ {0, 1}, g(b) = b; or (b) p(b) tJ_ {0, 1}, g(b) tJ_ {a, b} . 3. p(a) tJ_ {0, 1} and g(a) =I a, with p(b) tJ_ {0, 1}, g(b) =I b and p(a) =f p(b). 27 Cases 1, 2, and 3 set out in Theorem 3.10 can be rephrased in terms of whether a E L , p(a) E L, bEL or p(b) E L. Throughout the remainder of this section, we may assume a =I= b. The proof of Theorem 3.10 requires the construction of clopen sets A and B, as follows. Lemma 3.11. If {a , p(a)} n L =I= 0 and {b,p(b)} n L =I= 0 then X can be partitioned as Partition 1, that is, where each set is clopen and where a and b are in distinct sets in this partition. Proof. We construct Z, W and L as described in Lemmas 3.5 to 3.9 starting with the assumption a, b tf. Za U W 0 , so a, b tf. Z U W . If a E L then p(a) E p(L) ~ W. Since L n W =I= 0, the set {a,p(a)} n L has exactly one element. Similarly, {b,p(b)} n L has exactly one element. Let c E {a ,p(a)} n Land dE {b,p(b)} n L. There exist disjoint clopen sets A ~ L, and B ~ L with c E A and dE B. By Lemma 3.8, L n (Z U W) = 0. Then since A ~ Land B ~ L, we have An Z = 0, A n W = 0, B n Z = 0 and B n W = 0. Let K = L\(A U B), then p- 1 (L) = p- 1 (A) U p- 1 (B) U p- 1 (K). The sets A, B and K satisfy the following: 1. p(A) ~ vV; 2. g(A) ~ Z; 3. p(B) ~ W; 4. g(B) ~ Z; 5. p- 1 (A) n p - 1 (B) = 0; 28 6. p- 1 (A) s;;; p- 1 (L) and p- 1 (B) s;;; p- 1 (L); 7. p(p- 1 (A)) s;;; A and p(p- 1 (B)) s;;; B; 8. g(p- 1 (A)) s;;; p- 1 (A) and g(p- 1 (B)) s;;; p- 1 (B); 9. g- 1 (A) = 0 and g- 1 (B) = 0; 10. p - 1 (Z) n (AU B)= 0; 11. p(K) s;;; Wand g(K) s;;; Z; 12. p(p- 1 (K)) s;;; K and g(p- 1 (K)) s;;; p- 1 (K); 13. g- 1 (K) = 0. The proof of these thirteen claims now follows . As A s;;; L = p- 1 (W)\(Z U W) implies A s;;; p- 1 (W), it follows that p(A) s;;; W. Also, A s;;; p- 1 (W) = g- 1 (Z) implies A s;;; g- 1 (Z). Thus g(A) s;;; Z. Similarly, p(B) s;;; W and g(B) s;;; Z. Since A n B = 0, Lemma 3.3 gives p- 1 (A) n p- 1 (B) = 0. Given A s;;; L and 1 B s;;; L , it follows that p- 1 (A) s;;; p- 1 (L) and p- 1 (B) s;;; p- 1 (L). Then As p- (A) s;;; 1 p- 1 (A) we have p(p- 1 (A)) s;;; A and by Lemma 3.2, g(p- 1 (A)) s;;; p- (A). Similarly, p(p- 1 (B)) s;;; B and g(p- 1 (B)) s;;; p- 1 (B). Now, if g- 1 (A) =f. 0, then there is an e E g- 1 (A) with g( e) E As;;; L. So g(e) E L = g- 1 (Z) \(Z U W), and g 2 (e) E Z. Then Quasi-equation 3.2 implies g(e) E Z, which is a contradiction, since Z n L = 0. Similarly, we get g- (B) = 0. 1 1 By Lemma 3.9, p - 1 (Z) n L = 0, and with A s;;; L and B s;;; L, then p - (Z) n (AU B) = 0, and further, p- 1 (Z) satisfies all of the properties of Lemma 3.9. 1 Using K s;;; p- 1 (W) = g- 1 (Z) gives K s;;; p- 1 (W) and K s;;; g- (Z). Then it 1 follows that p(K) s;;; Wand g(K) s;;; Z. As p- 1 (K ) s;;; p- 1 (K), then p(p- (K)) s;;; K 29 and by Lemma 3.2, g(p- 1 (K)) ~ p- 1 (K). With K ~ L, by Lemma 3.8, g- 1 (£) = 0, and then we get g- 1 (K) = 0. The element a is either in A or in p- 1 (A), and the element b is either in Borin p- 1 (B) and these are all pairwise disjoint as required. 0 We have partitioned X into clopen sets as shown in Figure 3.5 and we can now complete the proof of Theorem 3.10. Recall that Theorem 3.10 states the following: Theorem 3.10. Assume a, bE X with {a, b} disjoint from {0, 1} and a =1- b, and a, b satisfy one of 1, 2, or 3, below. Then there exist disjoint clop en sets, A, B, Z, Wand L, such that Z, Wand L satisfy the properties of Lemma 3.6 to Lemma 3.9, and X can be partitioned as where the partition separates a, b, 0 and 1, and the sets satisfy the quasi-equations ofM. 1. p(a) = 1 and g(a) =1- a, with either (a) p(b) = 1 and g(b) =f. b; or (b) p(b) ~ {0, 1, a}, g(b) = b; or (c) p(b) ~ {0, 1, a}, g(b) =1- b. 2. p(a) ~ {0, 1}, g(a) =a and p(a) =1- p(b), with either (a) p(b) ~ {0 , 1}, g(b) = b; or (b) p(b) tj:; {0, 1}, g(b) tj:; {a,b}. 3. p(a) tf:; {0, 1} and g(a) =1- a, with p(b) tj:; {0, 1}, g(b) =f. band p(a) =f. p(b). 30 Pr-oof. Recall that W and Z may be chosen so that 1 E W and 0 E Z. For c E {a,b,p(a),p(b)}, with c i= 1, then c ~ W, and ford E {a,b,g(a),g(b)} with d i= 0, then d ~ Z. We have assumed a, b ~ {0, 1} and a i= b. The proof of Theorem 3.10 relies on showing {a,p(a)} and {b,p(b)} intersect L. Then Lemma 3.11 may be used. In part 1(a), we have p(a) = 1, by Quasi-equation 3.7 g(a) = 0, so g(a) i= a. Then p(a) = 1 implies p(a) E W and a E p- 1 (W), so a E L. Similarly, if p(b) = 1, and g(b) i= b, we have bE L. In parts 1(b) and 1(c) when p(a) = 1 and b ~ {0, 1}, we have a E L. But b ~ L since if bEL, then we have bE p - 1 (W), giving p(b) E W, which is a contradiction. Now p2 (b) = 1 and 1 E W, so this implies p(b) E p- 1 (1) ~ p- 1 (W), which implies p(b) E p- 1 (W)\(ZUW) = L. If g(b) =bas in part 1(b), or g(b) i= bas in part 1(c), with b E p- 1 (£), we have g(b) E g(p- 1 (L)) ~ p- 1 (L) by Lemma 3.8. Then a E L and p(b) E L. In parts 2(a) and 2(b), ifp(a) = p(b), by Quasi-equation 3.9, g(a) = g(b) . But this implies a= g(a) = g(b) = b, a contradiction. Therefore p(a) i= p(b). With g(a) =a, and g(b) = b in part 2(a), or g(b) i= b in part 2(b), we choose a E p- 1 (A) and bE p- 1 (B). By Lemma 3.11, p- 1 (A) ~ p- 1 (£) and p- 1 (B) ~ p - 1 (£). Accordingly, p(a) E Land p(b) E L. In part 3, we have p(a) i= p(b) as in part 2, and we again choose a E p- 1 (A) and b E p- 1 (B). Then p(a) ELand p(b) ELand Lemma 3.11 may be applied. Accordingly, X can be partitioned into disjoint clopent sets in which particular points are separated, such that, and the clopen sets satisfy the quasi-equational theory of M. 31 0 3.2 Partition 2 This section defines the second partition of X into clopen sets, which is illustrated in Figure 3.6. Figure 3.6: Partition 2 Theorem 3 .12 . For a, b E X with {a, b} disjoint from {0, 1} and a i- b, when p(a) ¢:_ {0, 1}, g(a) = a, p(b) ¢:_ {0, 1}, g(b) = b and p(a) = p(b), there exist disjoint clopen sets Z, W, L and R, such that Z, W and L satisfy the properties of Lemmas 3. 6 to Lemma 3. 9, and a, b ¢:_ Z U W, and X can be partitioned as Partition 2, that is, where these sets separate 0, 1, a and b and satisfy the quasi-equations of M . The proof of this Theorem requires the construction of a clopen set R as in the following Lemma. 32 1 Lemma 3.13. There exists a clopen set R ~ p- 1 (£) such that g(p- (L)) ~ R ~ p- 1 (£) and b tJ. R. Proof. By Lemma 3.8, g(p- 1 (£)) ~ p- 1 (£). If bE g(p- 1 (£)) then b = g(c), where c E p- 1 (£). Thus a= g(b) = g2 (c) = g(c) = b, a contradiction. Since X is a totally 1 disconnected Hausdorff space, there is a clopen set R with g(p- 1 (£)) ~ R ~ p- (£) and b tJ. R. 0 Lemma 3.14. If p(a) E L and p(b) E L, and g(a) =a, then X can be partitioned as Proof. We have disjoint clopen sets Z, W, Land p- 1 (Z) satisfying the properties of Lemma 3.6 to Lemma 3.9, specifically, p(L) ~ W, g(L) ~ Z, p(p- 1 (Z)) ~ Z and g(p- 1 (Z)) ~ p- 1 (Z). Further, the sets Rand p- 1 (£)\R satisfy the following: 1. p(R) ~ L; 2. g(R) ~ R; 3. g(p- 1 (L)\R) ~ R; 4. p(p- 1 (L)\R) ~ L. The set R ~ p- 1 (£), and it follows that p(R) ~ p(p- 1 (L)) ~ L, and g(R) ~ g(p- 1 (£) ~ g(p- 1 (L)\R) ~ R. Since g(p- 1 (£)) ~ R and p(p- 1 (£)) ~ L, it also follows that p(p- 1 (L)\R) ~ L. 0 We will now state the proof of Theorem 3.12. Recall that Theorem 3.12 states: Theorem 12 . For a, b E X with {a, b} disjoint from {0, 1} and a =/= b, when p(a) tJ. {0, 1}, g(a) = a, p(b) tJ. {0, 1}, g(b) = b and p(a) = p(b), there exist 33 disjoint clop en sets Z, W, L and R, such that Z, W and L satisfy the properties of Lemmas 3.6 to Lemma 3.9, and a, b ~ Z U W, and X can be partitioned as Partition 2, that is, where these sets separate 0, 1, a and band satisfy the quasi-equations of M . Proof. Given that g(b) = a, then pg(b) = p(a) and by Quasi-equation 3.5, pg(b) = p(b), so we have p(a) = p(b) . Now p2 (a) = 1 E W, so p(a) E p- 1 (W)\(Z U W) = L and a E p- 1 (£). As p(a) = p(b), it follows that bE p- 1 (£). Since a E p- 1 (£), then g(a) E g(p- 1 (£)) and with g(a) =a, then a E g(p- 1 (L) ~ R. If bE p- 1 (£), then g(b) E g(p- 1 (£), but g(b) =/= b, ~ R, and bE p- 1 (L)\R. Accordingly, p(a) E L and p(b) E L and Lemma 3.14 can be applied to obtain Partition 2 as a partition of X. Accordingly, X can be partitioned into disjoint clopen sets in which particular points are separated, such that X= Z U W u L (J p- 1 (Z) (J R (J (p- 1 (L)\(R). and the clopen sets satisfy the quasi-equational theory of M. 3.3 0 Partition 3 The third partition of clopen sets of X is constructed in this section and is illustrated in Figure 3.7. Theorem 3.15. For a, b E X with {a, b} disjoint from {0, 1} and a =/= b, when p(a) ~ {0, 1} , g(a) =!= a and p(b) ~ {0, 1}, g(b) =/= b and p(a) = p(b), there exists disjoint clopen sets Z, W, L, C, D and R, such that Z, W and L satisfy the 34 Figure 3. 7: Partition 3 properties of Lemma 3.6 to Lemma 3.9, and a, b tj. Z U W, and X can be partitioned as Partition 3, that is, X = Z U W U L U C U D U R U p- 1 (Z) where these sets separate 0, 1, a and band satisfy the quasi-equations of MI. The proof of this Theorem follows the construction of the clop en sets C and D. By Lemma 2.5, there are clopen sets C' and D' with a E C' , b E D' and C' n D' = 0. Then the case set out in Theorem 3.17 can be rephrased in terms of whether or not a E p- 1 (L) and b E p - 1 (L). Lemma 3 .16. If p(a) E L and p(b) E L, and g(a) f. a, then X can be partitioned as X= Z U W U L U C U D U R U p- 1 (Z). Proof. Choose the sets Z, W , Land p- 1 (Z) satisfying the properties of Lemma 3.6 to Lemma 3.9, more specifically, p(L) ~ W, g(L) ~ Z , p(p- 1 (Z)) ~ Z and g(p- 1 (Z)) ~ P- l(Z). 35 By Lemma 3.5, we may assume p(a) rf:. Z U W. This implies p(a) E L. Given that g(a) -::f-a, and g(a) E g(p- 1 (L)), then a rf:. g(p- 1 (L)) , since if a= g(a) with a E p- 1 (L), then we have p(a) E L. This gives a= g(a) E g(l) ~ Z, a contradiction. Accordingly, there exists a clopen set R ~ p- 1 (L) with g(p- 1 (L)) ~ a, b rf:. R. In addition, there exist disjoint clopen sets C and D defined as C = C' n p- 1 (L) and D = D' n p- 1 (L), with a E C and b E D, so C ~ p- 1 (L)\R and D = p- 1 (L)\(R U C). The sets C, D and R satisfy the following: (i) p(C) ~ p(D) ~ L; (ii) g(C) ~ g(p- 1 (L)) ~ Rand g(D) ~ g(p- 1 (L)) ~ R; (iii) p(R) ~ L; (iv) g(R) ~ g(p- 1 (L)) ~ R. Since C ~ p- 1 (L), p(C) ~ L. Then g(C) ~ g(p- 1 (L)) ~ R. Similarly, p(D) ~ L and g(D) ~ g(p- 1 (L)) ~ R. Now, R ~ p- 1 (L), so p(R) ~ p(p- 1 (L)) ~ L, and g(R) ~ g(p- 1 (L)) ~ R. D 'vVe can now proceed to the proof of Theorem 3.15. Proof. Given g(a) -::/- a and g(b) -::/- b and p(a) = p(b), Quasi-equation 3.9, implies that g(a) = g(b). We have disjoint clopen sets C and D, so we choose a E C and bE D. Since a E C ~ p- 1 (L), and ~ p- 1 (L), we have a E p- 1 (£), so p(a) E L and b E p- 1 (L). Therefore p(b) E L and Lemma 3.16 can be applied to obtain Partition 3 as a partition of X. Accordingly, X can be partitioned into disjoint clopen sets, in which particular points are separated, such that X= ZULU CUD U R U p - 1 (Z) 36 and t he clopen sets satisfy t he quasi-equational t heory of M . 3.4 0 Base Partition The base pa rt ition of X is illustrat ed in Figure 3.8. All cases not consider in previous part itions use this partition . F igure 3.8: Base Partition T h eorem 3 .17. For a, b E X and a -/= b, when a and b are as se t out in the following cases, X can be partitioned as f ollows: 1. a = 1, p(a) = 1, g(a) -/= a, with either (a) b = 0, p(b) = 1 and g(b) = b; or (b) b tf. {0, 1}, p(b) = 1, g(b) -/= b; or (c) b tf. {0, 1} , p( b) tf. {0, 1} , g (b) = b; or (d) b tf. {0, 1}, p(b) tf. {0, 1} , g(b) -/= b; or 37 (e) b tJ_ {0, 1}, p(b) = 0, g(b) =b. 2. a= 0, p(a) = 1, g(a) =a, and b tJ_ {0, 1}, with either (a) p(b) = 1, g(b) =I b; or (b) p(b) tJ_ {0,1} , g(b) = b; or (c) p(b) tJ_ {0,1}, g(b) =I b; or (d) p(b) = 0, g(b) =b . 3. a tJ_ {0,1} p(a) = 1, g(a) =I a and b tJ_ {0,1}, with either (a) p(b) tJ_ {0,1}, p(b) =a, g(b) = b; or (b) p(b) tJ_ {0, 1}, p(b) =a, g(b) =I b; or (c) p(b) = 0, g(b) = b. 4. a tJ_ {0,1} , p(a) tJ_ {0,1} and g(a) =a, with b tJ_ {0,1}, p(b) = 0 and g(b) = b; 5. a tJ_ {0 , 1}, p(a) tJ_ {0 ,1 } and g(a) =I a, with b tJ_ {0,1}, p(b) = 0, g(b) =band p(a) =I p(b); Moreover, Z, W and L, such that Z, W and L satisfy the properties of Lemma 3.6 to Lemma 3. 9, and a, b tJ_ Z U W, and these sets separate a and b and satisfy the quasi-equations of M . Proof. In each case, it is sufficient to show that a and b are in distinct sets in the collection Z, W, L, p - 1 (Z), p- 1 (L). In part 1 we have 1 =a E W. In part 1(a), we have 0 = b, so 0 E Z . In 1(b), 1(c) and 1(d) , b tJ_ {0,1} and by Lemma 3.2, we may assume b tJ_ W. In part 2, we have 0 =a E Z. For 2(a), 2(b) and 2(c), b tJ_ {0,1} so by Lemma 3.2 b tJ_ Z. 38 In part 3, we have a tJ. {0, 1} which implies a tJ. Z U W. We also have p(a) = 1 and 1 E W, so a E p - 1 (W). Then by Lemma 3.8, a E L. In part 3(a), we have b tJ. {0, 1} but p(b) = a E L, so p- 1 (p(b)) E p- 1 (£), which implies b E p- 1 (£). In 3(b), p(b) = 0 E Z implies p- 1 (p(b)) E p- 1 (Z), sob E p- 1 (Z). In part 4, a tJ. {0, 1} and p(a) tJ. {0, 1} but p(b) = 0. Therefore bE p- 1 (Z) and a tJ. p - 1 (Z). Similarly, in part 5, bE p- 1 (Z) but a tJ. p- 1 (Z). 0 3.5 The Proof of Theorem 3.1 In this section we complete the first proof of standardness, that is, the proof of Theorem 3.1. First, recall Theorem 3.1 states: Theorem 3.1 The structure M = (0, 1, 2; :F; T) where :F = {002, 110} and Tis the discrete topology, is standard. Proof. By Theorem 3.4, there are six possible conditions for a and for b giving rise to 36 possible cases. By symmetry, we need consider 15 situations where a and b come from distinct cases. Of the six situat ions where a and b satisfy the same property, two are immediately excluded because they imply a = b. A third situation also reduces to a = b. Thus there are 18 non-trivial situations to be considered. Recall the six possibilities for a: (i) a= 1; (ii) a= 0; (iii) a tJ. {0 , 1} ,p(a) = 1,g(a) f= a; (iv) {a,p(a)} disjoint from {0, 1},g(a) =a; 39 (v) {a,p(a)} disjoint from {0, 1},g(a)-/= a; (vi) a tf_ {0, 1}, p(a) = 0, g(a) =a. In Case (i) for a and case (ii), (iii), (iv), (v) or (vi) forb, these cases are covered by Theorem 3.17, in parts 1(a) - 1(e). In Case (ii) for a and case (iii), (iv), (v) or (vi) forb, this is the situation covered by parts 2(a)- 2(d) of Theorem 3.17. Now consider Case (iii) for a. Case (iii) forb is covered by part 1(a) of Theorem 3.10 and Case (vi) forb is covered by part 3(c) of Theorem 3.17. For Cases (iv) and (v) for b, there are two subcases that are independent of whether g(b) = b or g(b) -/=b. If p(b) -/=a, Cases (iv) and (v) forb are covered by part 1(b) and (c) of Theorem 3.10, and if p(b) = a, then these cases are covered by parts 3(a) and (b) of Theorem 3.17. Case (iv) for a and Case (iv) forb also has two subcases independent of whether g(b) = b or g(b) -/= b. The first subcase, when p(a) -/= p(b) is covered by part 2(a) of Theorem 3.10 and the second subcase when p(a) = p(b) is covered by Theorem 3.12. Then Case (v) forb is covered by part 2(b) of Theorem 3.10 and Case (vi) for b is covered by part 4 of Theorem 3.17. Similarly, in Case (v) for a and Case (v) forb, when p(a)-/= p(b), then this case is covered by part 3 of Theorem 3.10, and when p(a) = p(b), this is covered by Theorem 3.15. Then Case (vi) forb is covered by part 5 of Theorem 3.17. Finally, consider Case (vi) for a and case (vi) for b. With 0 = p(a) = p(b), we have g(a) = g(b), which gives a= b, and this case is not possible. 0 Given the algebra M = ( {0, 1, 2}; F; T) where F = {002, 110} and 'T is the discrete topology, there is a finite cover of disjoint clopen sets in which the particular points a and b are separated, that satisfies the quasi-equational theory of M. Therefore, M is standard. 40 Chapter 4 Example 2 The second example that we show is standard is the structure Mlz = ({0, 1, 2}; F; T) where F = {001, 002} and T is the discrete topology. When g denotes the operation 002 and r denotes the operation 001, the structure M 2 is shown in Figure 4.1. {v 0 0 ~ 0 9 ~ 1 Figure 4.1: The Structure of Mz Applying g and r to the 27 elements of M 2 3 yields Figure 4.2 The second result on standardness of unary algebras is the following theorem. Theorem 4.1. The structure M 2 = (0, 1, 2; F; T) where F = {001, 002} and T is the discrete topology, is standard. The proof of Theorem 4.1 uses Theorem 2.4 to show that M 2 is standard by showing that for each X E Mod 7 Thqe (M) 2 , and each pair a, b E X with a =1- b 41 112 (201) ~ 212 221 )r r( ~ 121 110 _;) r 9 211 Figure 4.2: The Structure of M 2 3 there is a partition of clopen sets that separates a and b such that the blocks of the partition satisfy the quasi-equations of M 2 . Note that M 2 has one constant valued function, 0. The equations and quasi-equations satisfied by M 2 include (4.1) ~ gn(x) ~ g(x), n > 1 (4.2) gnr(x) ~ 0, n > 0 (4.3) rg(x) ~ r(x) (4.4) rng(x) ~ 0, n > 1 (4.5) 42 r(x) ~ 0 ~ g(x) ~ 0 (4. 6) g(x) ~ g(y) (4.7) ~ r(x) ~ r(y) (4.8) g(x) ~ r(y) =? g(x) ~ 0 That t hese quasi-equations hold can be seen from Table 4. 1. r g r2 92 rg gr rg2 r2g g2r grz 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 1 2 1 1 0 0 0 2 0 X Table 4.1: Operations of M2 Througho ut the remainder of Chapter 4, X is a Boolean model of the quasiequational t heory of M 2 . The proof of T heorem 4.1 requires Lemma 2.5, Lemma 2.6 and Lemma 2.7, together with the following Lemmas which provide useful properties of subsets of X.: Lemma 4 .2 . Assume A, B ~ X. (i) If A ~ r - 1 (B), then g(A) ~ r - 1 (B); (ii) If A ~ g- 1 (B), then g(A) ~ g- 1 (B); (iii) If ~ r- 1 (B) and 0 E B, then r(A) ~ r - 1 (B); (iv) If A ~ g- 1 (B) and 0 E B, then r(A) ~ g- 1 (B). Proof. For property (i), assume r(A) = rg(A) so rg(A) ~ ~ r - 1 (B), so r(A) ~ By Quasi-equation 4.4, This implies r- 1 (rg(A)) ~ r- 1 (B), and g(A) ~ r - 1 (A) . 43 Similarly, for property (ii), if ~ g- 1 (B) then g(A) ~ Band by Quasi-equation 4.2, g2 (A) ~ Band g(A) ~ g- 1 (B). Now for property (iii), assume A S: r- 1 (B) and 0 E B. By Quasi-equation 4.1, rn(x) ~ 0. Then X= r- 2 (B) and AS: r- 2 (B). It then follows that r(A) S: r- 1 (B) . Finally, for property (iv), if A ~ g- 1 (B) and 0 E B, by Quasi-equation 3.4, we have gr(x) ~ 0, so X = r- 1g- 1 (0). Then A S: r- 1 g- 1 (B), which implies r(A) S: g- 1 (B) . 0 Lemma 4 .3. Assume A, B E X and An B = 0. Then r- 1 (A) n r- 1 (B) = 0. Proof. Let d E r - 1 (A) n r - 1 (B). Then r(d) E A and r(d) E B, but An B = 0, a contradiction. Therefore r - 1 (A) n r- 1 (B) = 0. 0 For each a E X, there are four possibilities and these are described in the next Theorem. Theorem 4 .4 . For each a E X, a Boolean model of the quasi-equational theory of M2, one of the following cases holds. (i) a= 0, r(a) = 0, g(a) = 0; ( ii) a =f. 0, r (a) = 0, g (a) = 0 i (iii) a =f. 0, r(a) =f. 0, g(a) = a; (iv) a =f. 0, r(a) =f. 0, g(a) =f. 0, g(a) =f. a. Proof. If a = 0, then Case (i) holds. Now assume a =/= 0. By Quasi-equation 4.1, r 2 (a) = 0, so it is sufficient to consider the cases when r(a) = 0 or r(a) =/= 0. If r(a) = 0, then Quasi-equation 4.6 gives g(a) = 0, which is Case (ii). Now assume r(a) =/= 0. By Quasi-equation 4. 6, g(a) =/= 0. Then by Quasi-equation 4.2, 44 g 2 (a) = g(a), so it is now sufficient to consider the cases when g(a) =a or g(a) f. a. The former condition gives Case (iii) and the latter gives Case (iv) . 0 Based on the four cases in Theorem 4.4, for a, b E X and a f. b, there are potentially 16 cases. By symmetry, we need consider 9 cases that can be described by one of four partitions. The possible partitions are: Partition 1 : X= Z U AU B (J K (J r- 1 (A) (J r- 1 (B) (J r- 1 (K) Partition 2 : X= Z (J L (J R (J r- 1 (L)\R Partition 3 : X=ZULUCUDUR Partition 4 : X=ZULur- 1 (L) Each of the 9 cases is covered by one of the four partitions. The cases and their partitions are shown in Table 4.2 - b 2 4 1 - - 3 4 1,4 1 - - - a 1 2 3 1 4 - 4 4 1,4 1,2 1,3 Table 4.2: Four cases for a and b give rise to 9 situations (13 including subcases) using four partitions. The following lemmas show the construction of clopen sets around the constant valued function 0. The illustration of these sets is shown in Figure 4.3. The partition illustrated in Figure 4.3 underlies all other partitions constructed in this Section. 45 Figure 4.3: Clopen Sets Around 0 Given b-=/: 0, by Lemma 2.5 there are clopen sets U and V, with 0 E U, b E V, and U n V = 0. Construct Z 1 clopen with 0 E Z1 and b tf. Z1. If there is an element a E X with a -=/: 0, we may assume {a, b} and Z 1 are disjoint. Now define the set Z2 as: Lemma 4.5. Z 2 satisfies r(Z2) <::;; Z2 and g(Z2) <::;; Z2. Proof. We have r(Z2) <::;; r(r - 1(Z1)), so r(Z2 ) <::;; Z 1 . Then since 0 E Z1, we have X = r - 2(0) <::;; r- 2(ZI), so Z 2 <::;; r- 2(Z1 ), that is, r(Z2) <::;; r- 1(Z1). By Quasi- 1 equation 4.3, gr(x) = 0 E Z 1. Therefore gr(Z2) <::;; Z1, so r(Z2) <::;; g- (ZI), and accordingly we have r(Z2) <::;; Z 2. Now g(Z2) <::;; g(g- 1 (Z1)) <::;; Z 1. Therefore, g(Z2) <::;; Z 1, and Z2 <::;; g- 1 (ZI). By part (ii) of Lemma 4.2 g(Z2) <::;; g- 1(Z1). By quasi-equation 4.4, rg(Z2 ) = r(Z2) <::;; Z2. Then g(Z2) <::;; r- 1 (Z2) <::;; r- 1 (Z1 ). Therefore g(Z2) <::;; Z2 . 46 0 Define the sets S and T as follows: S = r(r- 1 (Z2)\g- 1 (Z2)) T = g(r- 1 (Z2)\g- 1 (Z2)) Lemma 4.6. The sets S U T and g(g- 1 (Z2)) U r(g- 1 (Z2)) are disjoint. Proof. Assume a E S n r(g- 1 (Z2)). This implies a= r(b) with bE r- 1 (Z2)\g- 1 (Z2)· Thus, r(b) E Z2 and g(b) ~ Z2. However, a = r(c) for some c E g- 1 (Z2), that is, g(c) E Z 2 . Since r(b) = r(c), Quasi-equation 4.7 implies g(b) = g(c). This is a contradiction as g(b) ~ Z2 and g(c) E Z2 . Assume a E T n r(g- 1 (Z2 )). Now, a E T implies a = g(b) for some b E r- 1 (Z2)\g- 1 (Z2), with r(b) E Z2 and g(b) ~ Z2. But a= r(c) for some c E g- 1 (Z2), that is, g(c) E Z 2 and we have a = g(b) = r(c). Then Quasi-equation 4.8 implies g(b) = r(c) = 0. Since g(b) ~ Z2 , this is a contradiction. Assume a E S n g(g- 1 (Z2)). Again a E S implies a = r(b) with r(b) E Z2 and g(b) ~ Z2. But a= g(c) for some c E g- 1 (Z2) so g(c) E Z2 . By Quasi-equation 4.8 we have a= r(b) = g(c) = 0, a contradiction as this gives a= 0. Assume a E T n g(g- 1 (Z2 )). Again a E T implies a = g(b) for some b with r(b) E Z2 and g(b) ~ Z2. But a= g(c) with c E g- 1 (Z2), and g(c) E Z2. We have a= g(b) = g(c) with g(b) ~ Z 2 and g(c) E Z 2 , a contradiction. 0 Lemma 4 . 7. Claim g- 1 (Z2 ) ~ r- 1 (Z2 ) Proof. Let c E g- 1 (Z2). Then g(c) E Z2 and rg(c) E r(Z2) ~ Z2. By Quasi-equation 4.4, rg(c) = T(c) E r(Z2) s;;; Z2 . So r(c) E z2, which implies c E r - 1 (Z2)· 47 0 Since the sets referred to in Lemma 4.6 are all closed we may use Lemma 2.7 to find Z clopen such that Z <:;;; Z 2 ; Z and S U T are disjoint; and Note that 0 is in Z. Lemma 4.8. The following properties hold for Z: (i) r(Z) <:;;; Z; (ii) g(Z) <:;;; Z; and (iii) g- 1 (Z) = r - 1 (Z). Proof. Since Z <:;;; Z 2 and Z 2 <:;;; g - 1 (Z2 ) , we have r(Z) <:;;; r(g- 1 (Z2 )). But by choice of Z, the latter set is contained in Z. Similarly g(Z) <:;;; g(g- 1 (Z 2 )) and the choice of Z implies g(Z) <:;;; Z. Let c E g- 1 (Z). Then g(c) E Z and rg(c) E r(Z) <:;;; Z. By Quasi-equation 4.4, rg(c) = r(c) E r(Z) <:;;; Z. So r(c) E Z, which implies c E r - 1 (Z). Therefore, g- 1 (Z) <:;;; r- 1 (Z). Notice that g(g- 1 (Z2 )) <:;;; Z implies g- 1 (Z2 ) <:;;; g - 1 (Z) . However, Z <:;;; Z 2 implies g- 1 (Z) <:;;; g- 1 (Z2 ). Thus g- 1 (Z) = g- 1 (Z2). To show r- 1 (Z) <:;;; g- 1 (Z) assume there is an a E r- 1 (Z)\g- 1 (Z). As Z <:;;; Z 2 and by the previous statement, we have a E r - 1 (Z2 )\(g- 1 (Z2 ). Hence, r(a) E S which is disjoint from Z. But the choice of a gives r(a) E Z. This contradiction gives r - 1 (Z) <:;;; g - 1 (Z) and we have r - 1 (Z) = g - 1 (Z). D The behaviour of g and r on Z is illustrated in Figure 4.4. 48 z (:J0 Figure 4.4: The Element Z If X = Z then X has been partitioned by clopen sets as required by Theorem 2.4. Now assume Q = X\Z is non-empty. We need to construct more clopen sets that cover X-::/= Z. We may assume Z is chosen so that r- 1 (Z) = g- (Z). 1 Lemma 4.9. Let L = r- 1 (Z)\Z, then (i) g(L) ~ Z and r(L) ~ Z; (iii) g- 1 (L) = 0. Proof. We have L ~ r - 1 (Z) = g- 1 (Z) so L ~ r - 1 (Z) and L ~ g- 1 (Z). Then it 1 follows that r(L) ~ Z and g(L) ~ Z. As r- 1 (£) ~ r - 1 (£), by Lemma 4.2, g(r- (L)) ~ r- 1 (£). Now assume e E g- 1 (£). Then g(e) E L, so g(e) E r- 1 (Z) and g(e) ~ Z. But r- 1 (Z) = g- 1 (Z), which implies g(e) E g- 1 (Z) and g 2 (e) E Z. Quasi-equation 4.2 implies g(e) E Z, a contradiction. Therefore, g- 1 (L) = 0. Accordingly, there is a disjoint clopen set around the element 0, such that as previously illustrated in Figure 4.3. Corollary 4.10. For an element a, exactly one of the following hold. (i) a, r(a) E Z; or 49 0 (ii) a E L and r(a) E Z; or (iii) a E r- 1 (L) and r(a) E L. 4 .1 Partition 1 In this section the first non-trivial partition of X into clopen sets is constructed. This partition is larger than the base partition and is illustrated in Figure 4.5. Figure 4.5 : Partition 1 Theorem 4 .11. Assume a, b E X with {a, b} disjoint from {0} and a =f- b, and a, b satisfying one of 1, 2 or 3 below. Then there exist disjoint clopen sets, A, B, Z, L an d K, such that Z and L satisfy the properties of Lemmas 4.8 and 4.9, and X can be partitioned as, where the partition separates a, b and 0, and the sets satisfy the quasi- equations 1. r(a) = 0 and g(a) = 0, with either 50 (a) r(b) = 0 and g(b) = 0; or (b) r(b) rj. {0, a}, g(b) = b and r(a) =/= r(b); or (c) r(b) rj. {0, a}, g(b) rj. {0, b} and r(a) =J r(b). 2. r(a) =/= O,g(a) =J 0, g(a) =a and r(a) =J r(b), with either (a) r(b) =/= 0, g(b) = b; or (b) r(b) =J 0 and g(b) rj. {O,a,b}. 3. r(a) =/= 0, g(a) rj. {O,a}, with r(b) =J 0, g(b) rj. {O,b} and r(a) =/= r(b). Throughout the remainder of this section, we may assume a, b and 0 are distinct. Fix Z and L that satisfy Lemmas 4.8 to 4.9, such that a, b rj. Z. The proof of this Theorem requires the construction of clopen sets A and B below . The cases set out in Theorem 4.11 can be rephrased in terms of whether a E L, r(a) E L, bEL or r(b) E L. Lemma 4 .12. If {a, r(a)} n L =J 0, and {b, r(b)} n L =/= 0, then X can be partitioned as Partition 1, that is with r and g acting on the sets as illustrated in Figure 4.5 and a and b in distinct sets. Proof. Since a =J 0, a rj. Z, and by Corollary 4.10 either a E Lor r(a) E L, but not both, we let c E {a,r(a)} n Land let dE {b,r(b)} n L. There is a clopen set ~ L with c E A and d rj. A. There is also a clopen set B ~ L with d E B and c rj. B and An B = 0. By Lemma 4.9, L n Z = 0. Therefore, we have An Z = 0, and B n z = 0. 51 Let K = L\(A U B), then r- 1 (L) = r- 1 (A) U r- 1 (B) u r- 1 (K). The sets A, B and K satisfy the following: 1. r(A) ~ Z and g(A) ~ Z; 2. r(B) ~ Z and g(B) ~ Z; 3. r(K) ~ Z and g(K) ~ Z; 4. r- 1 (A), r- 1 (B) and r- 1 (K) are pairwise disjoint; 5. r- 1 (A) ~ r- 1 (L), r- 1 (B) ~ r- 1 (L) and r- 1 (K) ~ r- 1 (L); r(r- 1 (K)) ~ K; 6. r(r- 1 (A)) ~ A, r(r- 1 (B)) ~ 1 7. g(r- 1 (A)) ~ r- 1 (A), g(r- 1 (B)) ~ r- 1 (B) and g(r- 1 (K)) ~ r- (K); 8. g- 1 (A) = 0, g- 1 (B) = 0 and g- 1 (K) = 0. The proof that the above eight claims are true follows . As A ~ r- 1 (Z)\Z implies A ~ r - 1 (Z), it follows that r(A) ~ Z. Since A ~ r- 1 (Z) = g - 1 (Z), then A ~ g - 1 (Z) and g(A) ~ Z. Similarly, we have r(B) ~ Z, g(B) ~ Z, r(K) ~ Z and g(K) ~ Z. 1 Since A, B and K are pairwise disjoint, Lemma 4.3 gives r - 1 (A), r- (B) and r - 1 (K) are pairwise disjoint. Given A <;:;; L, B ~ L and K ~ L, it follows that r- 1 (A) ~ r - 1 (L) , r- 1 (B) ~ r- 1 (L) and r - 1 (K) ~ r- 1 (L). As r - 1 (A) ~ r - 1 (A), then r(r - 1 (A)) ~ 1 and by Lemma 4.2, g(r - 1 (A)) ~ r - (A). 1 Similarly, r(r- 1 (B)) ~ B, g(r- 1 (B)) ~ r - 1 (B), r(r - 1 (K)) ~ K, and g(r - (K)) ~ r - 1 (K). Now if g- 1 (A) =I= 0, then there is acE g- 1 (A) with g(c) E ~ L. So g(c) E L = g- 1 (Z)\Z and g 2 (c) E Z. Then by Quasi-equation 4.2 g(c) E Z, a contradiction since Z n L = 0. Therefore, g- 1 (A) = 0. Similarly, g- 1 (B) = 0 and g- (K) = 0. 1 52 0 We have partitioned X into clopen sets as shown in Figure 4.5 and can now complete the Proof of Theorem 4.11 Recall that Theorem 4.11 states: Theorem 4.11. Assume a, b E X with {a, b} disjoint from {0} and a =f. b, and a, b satisfying one of 1, 2 or 3 below. Then there exist disjoint clopen sets, A, B, Z, L and K, such that Z and L satisfy the properties of Lemmas 4.8 and 4.9, and X can be partitioned as, where the partition separates a, band 0, and the sets satisfy the quasi-equations of :Ml 2 . 1. r(a) = 0 and g(a) = 0, with either (a) r(b) = 0 and g(b) = 0; or (b) r(b) ¢:. {0, a}, g(b) =band r(a) =f. r(b); or (c) r(b) ¢:. {0, a}, g(b) ¢:. {0, b} and r(a) =f. r(b). 2. r(a) =f. 0, g(a) =f. 0, g(a) =a and r(a) =f. r(b), with either (a) r(b) =f. 0, g(b) = b; or (b) r(b) =f. 0 and g(b) ¢:. {0, a, b}. 3. r(a) =f. 0, g(a) ¢:. {0, a}, with r(b) =f. 0, g(b) ¢:. {0, b} and r(a) =f. r(b). Proof. Recall that Z may be chosen so that 0 E Z and for c E {a, b, r(a), r(b)}, with c =f. 0, then c ¢:. Z. We have assumed a =f. 0, b =f. 0 and a =f. b. The proof of 53 Theorem 4.11 relies on showing {a , r(a)} and {b, r(b)} intersect L so that Lemma 4.12 may be used. Since a i= 0 and b i= 0, we have a, b tf:. Z. When r(a) = 0, by Corollary 4.10, a E L as r(a) E Z. When r(a) i= 0, by assumption r(a) t/:. Z, so by Corollary 4.1 0, a E r- 1 (L) and r(a) E L. Similarly for b. In part 1, r(a) = 0, so a E L. The three subcases of part 1 rely on b. In part 1(a), r(b) = 0, sob E L. In both parts 1(b) and 1(c), we have r(b) i= 0, so by the above, r(b) E L . In part 2 and part 3, r(a) i= 0, so r(a) E L, and r(b) i= 0 so r(b) E L. Accordingly, X can be partitioned as By Lemma 4.12, the operations on the blocks behave as indicated in Figure 4.5. 4 .2 [] Partition 2 The second partition of X is constructed in this section and is illustrated in Figure 4.6. Theorem 4.13. For a, b E X with {a, b} disjoint from {0} and a i= b, when r(a) i= 0, g(a) =a, r(b) i= 0, g(b) = a and r(a) = r( b), there exist disjoint clopen sets, Z , L and R, such that Z and L satisfy the properties of Lemmas 4.8 to Lemma 4.9, and a, b t/:. Z, and X can be partitioned as, X= Z U L U R U r- 1 (L)\R where the sets separate a, b and 0 and the sets satisfy the quasi- equations of M 2 . 54 Figure 4.6: Partition 2 Throughout this section, assume g(a) = a and g(b) = a, and a, b 1. Z. The proof of this Theorem requires the construction of the clopen set R in the following Lemma. 1 Lemma 4 .14. There exists a clopen set R ~ r- 1 (L) such that a E g(r- (L)) ~ R and b 1. R. Proof. Given that g(a) = a and a 1. Z, then a E g(r- 1 (L)) and by Lemma 4.9, g(r- 1 (L)) ~ r- 1 (L) . If b = g(c) for some c, then g(b) = g2 (c), but g 2 (c) = g(c) by Quasi-equation 4 .2, so a = g(b) = g(c) = b, a contradiction, as a I= b. Thus b 1. g(r- 1 (L)). Since X is a totally disconnected Hausdorff space, there is a clopen set R with g(r - 1 (L)) ~ R ~ r - 1 (L) and bE r- 1 (L)\R. Therefore b ~ R. 0 Lemma 4.15 . If r(a) = r(b) and r(a), r(b) E L, then X can be partitioned as X= Z U L U R U (r- 1 (L)\(R). Proof. We have disjoint clopen sets Z, and L satisfying the properties of Lemmas 4.8 to Lemma 4.9, specifically, r(L) ~ Z, and g(L) ~ Z. Further, the sets Rand r - 1 (L)\R set out in Lemma 4.14 satisfy the following: 55 1. r(R) ~ L; 2. g(R) ~ R; 3. g(r- 1 (L)\R) ~ R; 4. r(r - 1 (L) \R) ~ L. 1 Since R ~ r- 1 (L), it follows that r(R) ~ r(r- 1 (L)) ~ L, and g(R) ~ g(r- (L) ~ R . It also follows that with g(r- 1 (L)) ~ r(r- 1 (L)) ~ L, then g(r- 1 (L)\R) ~ Rand r(r- 1 (L)\R) ~ L. 0 We have partitioned X into clopen sets as illustrated in Figure 4.6, and now show the proof of Theorem 4.13. Proof. Given g(b) =a, rg(b) = r(a) and by Quasi-equation 4.4, rg(b) = r(b), then r(a) = r(b). Now r 2 (a) = 0 E Z, so this implies r(a) E r- 1 (Z)\Z = L and a E r- 1 (L). With r(a) = r(b), we have b E r- 1 (£) and since a E r- 1 (£), this implies g(a) E g(r- 1 (L)). Then, given g(a) =a, we have a E g(r- 1 (£)) ~ R. If bE r- 1 (£), this implies g(b) E g(r- 1 (£)), but g(b) # b, so that means b ~ R , and bE r- (L)\R. 1 Accordingly, r(a) E L and r(b) E L and Lemma 4.15 can be applied to obtain Partition 2 as a partition of X. Therefore, X can be partitioned as : X= Z U L (J R (J (r - 1 (L)\(R) D 4.3 Partition 3 The third partition of X in this section is shown in Figure 4.7. 56 ~ C D R \I () L r cb g Figure 4.7: Partition 3 Theorem 4.16. For a, bE X with {a, b} disjoint from {0} and a I= b, when r(a) /= 0, g(a) rf. {0, a}, and r(b) I= 0, and g(b) rf. {0, b} and r(a) = r(b), there exists disjoint clopen sets Z, L, C, D and R, such that Z, and L satisfy the properties of Lemma 4.8 and Lemma 4.9, and X can be partitioned as, X = Z (J L (J C (J D U R. where these sets separate a, b and 0 and satisfy the quasi-equations of M2. The proof of this Theorem requires the construction of the clop en sets C and D, as follows. By Lemma 2.5, there are clopen sets C' and D' with a E C', b E D' and C' n D' = 0. Then the case set out in Theorem 4.16 can be rephrased in terms of whether a and b arc in r- 1 (L). Lemma 4.17. If r(a) E L and r(b) E L , and g(a) I= a and g(b) /= b, then X can be partitioned as X=ZULUCUDUR. 57 Proof. The sets Z and L satisfy the properties of Lemma 4.8 to Lemma 4.9, more specifically, r(L) ~ Z, and g(L) ~ Z. By Lemma 2.5, we may assume r(a) tJ. Z. This implies r(a) E L. Since g(a)-::/= a, and g(a) E g(r - 1 (L)), this implies a tJ. g(r- 1 (L)), because if a= g(c) with c E r- 1 (L), then g(a) = g2(c) = g(c), by Quasi-equation 4.2, that is g(a) =a, a contradiction. Similarly, b tJ. g(r- 1 (L)). Accordingly, there exists a clopen set R ~ r- 1 (L) with g(r- 1 (L)) ~ {a, b} disjoint from R. In addition, there exist disjoint clopen sets C and D defined as C = C' n r- 1 (L) and D = D' n r- 1 (L), with a E C and bE D. So we have C ~ r- 1 (L)\R and D ~ r - 1 (L)\(R U C). The sets C, D and R satisfy the following: 1. r(C) ~ r(D) ~ L ; 2. g(C) ~ g(r- 1 (L)) ~ Rand g(D) ~ g(r- 1 (L)) ~ R; 3. r(R) ~ L; 4. g(R) ~ g(r- 1 (L)) ~ R. Since C ~ r- 1 (L), then r(C) ~ L. This implies g(C) ~ g(r- 1 (L)) ~ R. Similarly, r(D) ~ g(D) ~ g(r- 1 (L)) ~ R. Now R ~ r- 1 (L), so r(R) ~ r(r- 1 (L)) ~ L, and g(R) ~ g(r- 1 (L)) ~ R. 0 We have partitioned X into clopen sets as shown in Figure 4.7 and can now complete the Proof of Theorem 4.16 Proof. Given that r(a) = r(b), we have g(a) = g(b) by Quasi-equation 4.7 with g(a) -::/= a and g(b) -::/= b. Now we have disjoint clopen sets C and D, so we choose a E C and bE D. Since a E C ~ r - 1 (L), and bED ~ r - 1 (L), we have a E r- 1 (L), so r(a) ELand bE r - 1 (L). Therefore bEL and Lemma 4.17 can again be applied. Accordingly, X can be partitioned as follows. 58 X=ZULUCUDUR 0 with a and b in distinct sets. 4.4 Base Partition The final partition for this example is constructed here and shown in Figure 4.8. Figure 4.8: Base Partition Theorem 4.18. For a, bE X with {a, b} disjoint from {0} and a=/= b, when a and b are as set out in the following cases, X can be partitioned as follows: 1. a= 0, r(a) = 0, g(a) = 0, with either (a) b =/= 0, r(b) = 0 and g(b) = 0; or (b) b =/= 0, r(b) =/= 0, g(b) =/= 0, g(b) = b; or (c) b =/= 0, r(b) =/= 0, g(b) ~ {0, b}. 59 2. a =f 0, r(a) = 0, g(a) = 0, r(b) = a with either (a) b =f 0, r(b) =f 0, and g(b) = b; or (b) b =f 0, r(b) =f 0, g(b) ~ {0, b} . Moreover, the disjoint clopen sets satisfy the properties of Lemma 4.8 and Lemma 4.9, and a, b ~ Z, and these sets separate a and b and satisfy the quasi-equations of M2. Proof. In each case, it is sufficient to show that a and b are in distinct sets in the collection Z, L, and r- 1 (L) . In part 1 we have 0 = a E Z and in 1(a), r(b) = 0 E Z, which implies bE r - 1 (Z). In part 1(b) and 1(c), b =f 0 and by Lemma 4.2, we may assume b ~ Z. In part 2, a =f 0, so a ~ Z, but r(a) = 0 E Z, so this implies a E r - 1 (Z) and by Lemma 4.9, we have a E L. Then we have b =f 0, but r(b) =a E L , so r- 1 (r(b)) E r - 1 (L), which gives b E r - 1 (L) . 4 .5 0 Proof of Theorem 4 .1 In this section we complete the second proof of standardness, t hat is, the proof of Theorem 4.1. Recall Theorem 4.1 states the following: Theorem 4.1 The structure M 2 = (0, 1, 2; F; T) where F = {001 , 002} and Tis the discrete topology, is standard. Proof. By Theorem 4.4, there are four possible conditions for a and forb giving rise to 16 possible cases. By symmetry, we need only consider 9 situations where a and b come from distinct cases . Recall the four possibilities fo r a: (i) a= 0, r(a) = 0, g(a) = 0; 60 (ii) a# 0, r(a) = 0, g(a) = 0; (iii) a# 0, r(a) # 0, g(a) =a; (iv) a# 0, r(a) # 0, g(a) tj. {0, a}. In Case (i) for a and Case (ii), (iii) or (iv) for b, these cases are covered by part 1(a), (b) and (c) of Theorem 4.18. In Case (ii) for a and Case (ii) for b, this is the situation covered by part 1(a) of Theorem 4.11. In Case (ii) for a and Case (iii) for b, there are two subcases. If r(b) = a, this case is covered by part 2(a) of Theorem 4.18, and if r(b) # a, this case is covered by part 1(b) of Theorem 4.11. Similarly in Case (ii) for a and Case (iv) for b, there are two sub cases. If r(b) = a, this case is covered by part 2(b) of Theorem 4.18, and if r(b) #a, this case is covered by part 1(c) of Theorem 4.11. In Case (iii) for a and Case (iii) for b, this situation is covered by part 2(a) of Theorem 4.11. In Case (iii) for a and Case (iv) forb there are two subcases. When g(b) = a, this case is covered by Theorem 4.13, and when g(b) # a, this case is covered by part 2(b) of Theorem 4.11. Finally, in Case (iv) for a and Case (iv) for b, there are two subcases. The case when r(a) = r(b) is covered by Theorem 4.16, and the case when r(a) # r(b) is covered by part 3 of Theorem 4.11. Each of the four partitions are isomorphic to subalgebras of M 2 , so they do satisfy the quasi-equations of M 2 . 0 Given the algebra M 2 = ({0, 1, 2}; F; T) where F = {001, 002} and T is the discrete topology, for any X E Mod 7 Thqe (M) 2 , and any pair of elements a, b E X, we have constructed a finite cover of clopen sets in which the particular points a and b are separated, and the clopen sets satisfy the quasi-equational theory of M 2 . Therefore, M 2 is standard. 61 The proof of Theorem 4.1 uses Theorem 2.4 to show that M2 is standard by showing that for each X E Mod 7 Thqe (M) 2 , and each pair a, b E X with a =/= b there is a partition of clopen sets that separates a and b such that the blocks of the partition satisfy the quasi-equations of M 2 . 62 Chapter 5 Two More Examples 5.1 Example 3 The third example for standardness is the structure Ivlh = ( {0, 1, 2}; F; T) where F = {110, 112} and Tis the discrete topology. When p denote the operation 110 and b denote the operation 112, the structure M 3 is shown in Figure 5.1. {D 0 /, 6 ~ b ~ ~ Figure 5.1: The Structure of M3 Applying p and b to the 27 elements of M 3 3 yields Figure 5.2. This algebra is isomorphic to M 2 = ({0, 1, 2}; F; T) where F = {001, 002}, and is therefore standard. 63 ( 120 200 022 ~ p( )p 001.:) \z1oo p b CE211 ~ b p ~ ~ 222--"---... 0 ~ 112}) 111 ~~ ~ 016~ ~ ) 202p \ 102 220 p h ~ ~ 020 Figure 5.2: The Structure of :Ml 3 3 5.2 Example 4 This is the fourth example in which it was intended to determine whether or not the particular structure described below was standard . This structure, however, does not have any constant valued functions. Accordingly, using the same method as that in the p revious three examples, was more difficult that what was expected and this example remains incomplete. What follows is the beginning of the work that was completed for this example. The fourth example is the structure :Ml4 = ( {0, 1, 2}; F; T) where F = {002, 112} and T is the discrete topology. When g denote the operation 002 and s denote the operation 112, the structure :Ml 4 is shown in Figure 5.3. Applying g and s to the 27 elements of :Ml4 3 yields Figure 5.4. 64 (jJ s +----g--- 8 Figure 5.3: The Structure of M4 s 120 ~~~ ~ 210 ~ ~ ~ ~ QJ2 _g_ 1B s cEJ0--18 g s @0g -28 s s s s c!J0==28 g s .2o2==:2B (!_} g s ~ cEJ2==:1B g s ~ Figure 5.4: The Structure of Ml Conjecture 5.0 .1. The structure M 4 = (0, 1, 2; F; T) where F = {002, 112} and T is the discrete topology, is standard. The proof that M 4 is standard would require the use of Theorem 2.4 and would show that if M 4 is standard there will be a finite cover of clopen sets of each Boolean model of the quasi-equational theory of M 4 in which a and b are separated. 65 The equations and quasi-equations satisfied by MI 4 3 include (1) gn(x) ~ g(x) , n ~ 1 (2) sn(x) ~ s(x), n ~ 1 (3) sg(x) ~ s(x) (4) gs(x) ~ g(x) (5) g(x) ~ g(y) ¢:=:? s(x) ~ s(y) (6) g(x) ~ s(y) => x ~ y (7) g(x) ~ x and s(x) ~ x, and g(y) ~ y and s(y) ~ y => x ~ y. That these quasi-equations hold can be seen from Table 5.1. X 0 1 2 g 0 0 2 s 1 1 2 g2 82 0 0 2 1 1 2 sg 1 1 2 gs 0 0 2 sg'l: s2g gs2 g2s 1 1 2 0 0 2 0 0 1 1 2 2 Table 5.1 : Operations of MI 4 Conjecture 5.0.2. For each a E X, a Boolean model of the quasi- equational theory of ({0, 1, 2}; g, b) where g = 002 and s = 112, one of the following cases holds. (i) g(a) = s(a) =a; (ii) g(a) =a, s(a) =/=a; (iii) g(a) =!=a, s(a) =a; (iv) g(a) =/=a, s(a) =/=a. 66 Proof. By Quasi-equation 1, g2 (a) = g(a) and by Quasi-equation 2, s 2 (a) = s(a), so we consider the cases where g(a) = a or g(a) -/= a with s(a) -/= a or s(a) = a. If g(a) = a, and s(a) = a, this is Quasi-equation 6 and we get Case i. Then if s(a) -/= a we get Case ii. The remaining two cases are when g(a) -/=a and we need only consider when s(a) = a or s(a) -/=a. The former condition gives Case iii and D the latter Case iv. The proof that M 4 is standard will frequently require the application of Lemmas 2.5 and 2.6 and the following Lemmas which provide useful properties of subsets of X. Lemma 5 . 1. Given A closed and b E X such that b tJ. A, there exist clopen sets U, V with U n V = f/J, A ~ U and b E V. Lemma 5.2. Assume A, BE X. (i) If ~ s- 1 (B), then g(A) ~ s- 1 (B) (ii) If ~ g- 1 (B), (iii) If ~ s- 1 (B), then s(A) ~ s- 1 (B) then g(A) ~ g- 1 (B) (iv) If As;;: g- 1 (B), then s(A) s;;: g - 1 (B). Proof. Assume A, B E X. If A ~ s- 1 (B), then s(A) ~ B. By quasi equation 3, s(A) = sg(A), so sg(A) ~ B, and g(A) ~ s- 1 (B). If A ~ g- 1 (B), then g(A) ~ B. Quasi-equation 1 says g(A) = g2 (A) so g2 (A) ~ B. Then g(A) ~ g- 1 (B). If A ~ s- 1 (B), s(A) ~ B. By quasi-equation 2, s(A) = s 2 (A) ~ B. Then s(A) ~ s- 1 (B). If A ~ g- 1 (B), then g(A) ~ B, and by quasi-equation 4, g(A) = gs(A) ~ B, so s(A) ~ g- 1 B. D Lemma 5.3 . Assume A, BE X and An B = f/J. Then s- 1 (A) n s- 1 (B) = f/J. 67 Proof. Let dE s - 1 (A) n s- 1 (B) . Then s(d) E A and s(d) E B , but An B = 0, a contradiction. Therefore s- 1 (A) n s - 1 (B) = 0. This ends the work completed for this example. 68 0 Chapter 6 Summary In this work I proved that Example 1 and Example 2 are standard. Example 3 is also standard. In order to complete the proof that Example 4 is standard requires some more work. Because there is no constant valued function for Example 4, the partitions were more complicated than expected. There may be other techniques used including the application of the fo llowing theorem. Theorem 6 .1. If M is a 3-element unary structure with V, IL or Il.J) as an iso-reduct then M is non-standard. The proof of this theorem uses an inverse limit technique . This is a technique that I have not learned. The algebras that are known to be non-standard using this technique are shown in Table 2.1. If all remaining structures are found to be standard, the following Conjecture would hold. Conjecture 6.1.1. If M is a 3-element unary structure without V, IL or j[)) as an iso-reduct then M is standard. 69 Bibliography [1] Stanley Burris and H.P. Sankappanavar, A course in universal algebra, millenium ed., Springer, New York, Heidelberg, Berlin, 1981. [2] David M. Clark and Brian A. Davey, Natural dualities for the working algebraist, first ed., Cambridge University Press, Cambridge UK, 1998. [3] David M. Clark, Brian A. Davey, M. Haviar, J.G. Pitkethly, and M.R. Talukder, Standard topological quasi-varieties, Houston Journal of Mathematics 29 (2003), no. 4. [4] D.M. Clark, B.A. Davey, and J.G. Pitkethly, The complexity of dualisability: Three-element unary algebras, International Journal of Algebra and Computa- tion 13 (2003), no. 3. [5] J. Hyndman and J.G . Pitkethly, Standard unary algebras, Manuscript, 2001. 70