ALGEBRAS, LATTICES, VARIETIES VOLUME II

Transcription

1 ALGEBRAS, LATTICES, VARIETIES VOLUME II Ralph Freese University of Hawai i Ralph N. McKenzie Vanderbilt University George F. McNulty University of South Carolina Walter F. Taylor University of Colorado

2 Contents Chapter 10 Part of Chapter Taylor classes of varieties 1 Chapter 11 Abstract Clone Theory Abstract Clones and their Homomorphisms Representations of Clones Presentations of Clones Clone Homomorphisms and Concrete Functors Products and CoProducts of Clones [k] th Power Varieties Category-equivalence of varieties The interpretability lattice Hypervarieties Definable classes of varieties History and references for Chapter Index 177 vi

3 C H A P T E R T E N Part of Chapter Taylor classes of varieties Let M and N be two n n arrays, or matrices, of terms in a signature ρ, and F, G two n-ary terms in ρ. The (formal) matrix equation F (M) G(N) is taken as shorthand for the following sequence of n equations: F (M i,0,..., M i,n 1 ) G(N i,0,..., N i,n 1 ), (10.1) where i ranges from 0 to n 1. By the i th row of M, we mean the tuple M i,0,..., M i,n 1 ; thus Equation (10.1) can be summarized as F (i th row of M) G(i th row of N). Until further notice, we restrict this notation to the special case that F = G and the matrix entries M ij and N ij are variables. Notice I don t define a Taylor set of equations. This definitely is an array; in particular, repetition of rows can be important. Therefore it is a sequence, not a set. I don t propose to mention this particularly, but it;s good to write it that way, and to keep it in mind. DEFINITION Let F be an n-ary operations symbol, and M and N be n n matrices of variables, with x s along the diagonal of M and y s along the diagonal of N. The (M, N, F )-Taylor equations are F (x,..., x) x; F (M) F (N). (10.2) (This is an (n + 1)-tuple of equations.) If a variety V satisfies these equations, we may say that F is an (M, N)-Taylor term for V. A Taylor term for V is an (M, N)-Taylor term for V for some appropriate M and N. A Taylor term for an algebra A is a Taylor term for HSPA. 1

4 2 Chapter 10 Part of Chapter 10 We first note that we obviously can admit M and N under the apparently weaker condition that M i,i N i,i for each i < n. (For [C. Bergman], this is the definition.) For such M and N an easy change of variables on the i th row renders M i,i = x and N i,i = y. In fact we can utilize M and N under the even weaker conditions that M and N each have n columns (and any number of rows) and that ( j) ( i(j)) M i(j),j N i(j),j. (10.3) Under this condition we may simply build new (n n) matrices M and N where the i(j) th row of M is used as the j th row of M (and similarly for N and N ). On the other hand, if (10.3) fails for M and N, i.e. if there exists j such that M and N agree on their j th columns, then (10.2) is satisfied for F equal to x j, the j th coordinate projection. In other words if we allowed this situation every variety would have a Taylor term. To exclude that possibility, we require condition (10.3) to hold, at the bare minimum. We often deal with M and N that have been revised (as described above) so that {M jj, N jj } = {x, y} for all j. It is now apparent that for any two fixed appropriate n n matrices M, N, the existence of an (M, N)-Taylor term on V is a (strong) Mal tsev condition on V, as described at the start of This may be called the (M, N)-Taylor condition. As noted there, the validity of the (M, N)- Taylor condition for V is equivalent to the interpretability of the equations (10.2) in V. The major result of see Theorem 10.7 below creates two matrices of x s and y s, M and N, such that the (M, N)- Taylor condition is the weakest of all idempotent Mal tsev conditions. This means that if W is a finitely based idempotent non-trivial variety of finite signature, then (for these matrices M and N) the equations (10.2) are interpretable in W. EXAMPLES OF TAYLOR TERMS 1. Congruence permutability. It is not hard to see, in practice, that many of the Mal tsev conditions discovered in this chapter are either Taylor conditions or closely related to such conditions. We present one example now, while reserving the full statement to Theorem Consider the equations F (M) = F x x y x x y F y y y y y y = F (N). y x x y y y It is easy to check that each equation here is one of Mal tsev s original two equations for congruence permutability, and that both of those equations are included. Therefore the Taylor condition here is equivalent to the original condition of Mal tsev. (The second rows of M and N repeat

5 10.11 Taylor classes of varieties 3 the first rows; this is allowed by our definition.) Thus every congruencepermutable variety has this Taylor term. One may make a very similar construction of M and N for the strong Mal tsev condition of an n-ary near-unanimity term. (Page 266). Recall that the defining equations are F (x, y, y,..., y) y F (y, x, y,..., y) y F (y, y,..., y, x) y, for a single n-ary operation symbol F. It is not hard to see that these n equations can be packaged in a single n n matrix of equations, so that the matrix equation, together with idempotence, defines near-unanimity. Thus n-fold near unanimity is both a Mal tsev condition and a Taylor condition. 2. Congruence 3-permutability. Let V be a congruence-3-permutable variety. By Theorem 10.1 there are ternary V-terms p and q that satisfy these equations in V: x p(x, z, z); p(x, x, z) q(x, z, z); q(x, x, z) z. (10.4) We will see that F (x 0,..., x 8 ) = p(q(x 0, x 1, x 2 ), q(x 3, x 4, x 5 ), q(x 6, x 7, x 8 )) (10.5) is a Taylor term for V. We claim that F is an (M, N)-Taylor term with M and N taken as the following two matrices: x x x y y y y y y M = x x x x x x y y y ; x x y x x y x x y x x x x x x x x x N = x y y x y y x y y. y y y y y y y y y We first ask the reader (see Exercise 2) to verify that the matrices M and N satisfy (10.3). This establishes that M and N do form a Taylor condition. It is not hard to convert M (resp. N) to a 9 9 matrix that has x s (resp. y s) on the diagonal, in the manner described above. This conversion, however, is not necessary for establishing that F is a Taylor term for V. It is a simple matter (see Exercise 2) to verify that the equations F (M) F (N) follow from (10.5) and the 3-permutability equations (10.4). Thus the F defined in (10.5) is an (M, N)-Taylor term for V. Thus every 3-permutable variety has a Taylor term. The method described here for 3-permutability carries over with minimal changes to k-permutability for arbitrary finite k. Formula (10.5).

6 4 Chapter 10 Part of Chapter 10 defining F is a simple instance of the -operator which we are about to describe; namely, F = p q. Similar naïve methods can be applied to many of the Mal tsev conditions in this chapter, to show that they also yield Taylor conditions. The theorem that follows (Theorem 10.2) assures us that this is possible for any non-trivial idempotent Mal tsev condition. Non-trivial idempotent varieties Let us call an algebra trite if it has more than one element and each of its operations is a projection operation. One easily sees that every term operation in a trite algebra is a projection as well. A variety V will be called trivial or passive (see [C. Bergman]) if V has a trite model. (In this case, V has trite models of every cardinality.) If V has no trite models, then we call V non-trivial or active. One easily sees that the varieties of interest in this chapter with permutable, distributive, modular, regular, etc., congruences, and other classes as well never have trite models, and hence are active varieties. This observation is the starting point for the following theorem. The theorem was stated in Taylor [1977?]. THEOREM V has a Taylor term if and only if V has a reduct that is active and idempotent. In particular, if V has a Taylor term, then V is active. Proof. ( ) First let us suppose that V has a Taylor term; more precisely, that there exist an idempotent n-ary term F and square matrices M (resp. N) with x s (resp. y s) on the diagonal, such that V satisfies F (M) F (N). We will prove by contradiction that V has no trite models. So let A be a trite algebra in V; clearly A satisfies F (x 0,..., x n 1 ) x j for some j. Now we have F (j th row of M) F (j th row of N) as one identity of V. But this equation is synonymous with x y, which contradicts the assumption that A has more than one element. Thus V has no trite models, and hence is active. Clearly the F -reduct of V is active and idempotent. ( linear identities) For the converse, it will suffice to assume that V is idempotent and active, and to prove that V has a Taylor term. We will first give the proof for a special case: V has finite signature and is defined by a finite set of linear equations (page 302, before Theorem 10.25). This special argument requires little more than a systematic version of Example 2 above; it is of some interest in this chapter, because many of the Mal tsev conditions of the chapter are defined by linear equations. In this case, the method is so simple that for many of the smaller Mal tsev conditions of the chapter a Taylor condition can be computed by hand. Later we shall return to the general case. If σ and τ are terms for a given signature, then σ τ denotes the term that is defined by the equation σ τ(x 0,..., x rs 1 ) = σ(τ(x 0,..., x s 1 ),..., τ(x (r 1)s,..., x rs 1 )),

7 10.11 Taylor classes of varieties 5 with the right-hand side denoting substitution, (See XX reference for substitution still needed ). The reader may easily check that this multiplication is an associative operation on the set of all terms. Supposing the operations of V to be F 0, F 1,... F m 1, we define H to be the term F 0 F 1 F m 1. We claim H is a Taylor term for V. Let us assume that all F i have the same arity, N. (This assumption simplifies the notation, but does not essentially change any reasoning.) We may now use idempotence to derive two simple facts about the values of H on tuples that contain some special kinds of repetition. Note that the arity of H is N m. LEMMA Let s and t be idempotent terms in a variety V. (i). Suppose that s has arity N and t has arity N r for some r. If X is an input N r+1 -vector of variables, and if X has period N r, then V satisfies s t(x) t(x ), where X is one period, of length N r. (ii). Suppose that t has arity N and s has arity N r for some r. If X is an input N r+1 -vector of variables, and if X consists of N r constant blocks of length N, then V satisfies s t(x) s(x ), where X is formed by collapsing each constant block of length N. Proof. See Exercise 3. and H are idempotent terms as de- COROLLARY. If F 0,..., F m 1 scribed above, then for each i, V = F i (x 0,..., x N 1 ) H(X i ), (10.6) where X i is the sequence of variables x j, whose subscripts appear as the i th digit in the base-n expansion of the sequence 0, 1,..., N m 1. Now let us consider the set Σ of equations defining V. By assumption, each equation in Σ has the form F j (x σ(0),..., x σ(n 1) ) F k (x τ(0),..., x τ(n 1) ), (10.7) where σ and τ are selfmaps of {x 0,..., x N 1 }, and where j and k may be the same or different elements of {0,....m 1}. Now applying the substitution σ to Equation (10.6) we obtain V = F j (x σ(0),..., x σ(n 1) ) H(σX j ), with X j as indicated after (10.6), and σx j the result of applying the substitution σ to the sequence X j. (It is equal to σ X j.) By transitivity of, we now have V = H(σX j ) H(τX k ) (10.8) (one such equation for each linear V-identity of type (10.7)). Now, to show that H is a Taylor term for V, we will show that condition (10.3) holds

8 6 Chapter 10 Part of Chapter 10 for the matrices M and N whose rows come from the (left and right sides of) Equations (10.8). We prove (10.3) by contradiction. If it fails then there exists q {0,..., N m 1} such that M i,q = N i,q for all i. In terms of our matrices M and N, this means that for each Equation (10.8), the two sides of (10.8) have the same variable in the q th position of H. This means that if we define H : A M n A via H(x 0, x 1,... ) = x q (the q th projection operation), then the algebra A, H satisfies each Equation (10.8). Now let us augment the algebra A, H with operations F i defined as follows. Expand q in base-n notation, and let d i be the i th digit in this expansion. Then define F i : A N A to be the d th i projection operation. It is not hard to check directly from the definitions that the algebra A; H, F i i<m models Equation (10.6). We wish to prove that the trite algebra A; H, F i, i<m models Equations (10.7), in other words that its F i -reduct lies in V. So we suppose that a 0, a 1,... A, and we compute F j (a σ(0),..., a σ(n 1) ) = H(σA j ) = H(τA k ) = F k (a τ(0),..., a τ(n 1) ). (Here A i is defined analogously to the X i found in (10.6): it is the sequence of length N m of elements of {a 0,..., a N 1 } whose subscripts appear as the i th digit in the base-n expansion of the sequence 0, 1,..., N r 1.) Here the first and last equalities come from (10.6), and the middle equality comes from (10.8). Thus V has the trite model A; F i i<n, in contradiction to our assumption that V is an active variety. Thus H is an (M, N)-Taylor term for V, where M and N arise from Equations (10.8) above. ( general case.) In fact it will be enough to prove this implication in the special case that V is an idempotent variety of finite signature, defined by a finite set of equational axioms. For, given an active variety V = Mod Σ, with Σ possibly infinite, we may first apply the Compactness Theorem 7.30 to prove that there is a finite subset Σ 0 of Σ such that Mod Σ 0 is active. (Details are left to the reader in Exercise 4.) So now we assume, as before, that V has the signature of operation symbols F 0,..., F m 1, each N-ary, and that V is defined by a finite set Σ of equations in the operations F i. The equations in Σ are taken to be µ j ν j for 0 j < k. Depth. The depth h(σ) of a term σ (in any signature ρ) is defined recursively as follows. Each variable has depth 0. If σ = F (τ 0,..., τ n 1 ) for an n-ary ρ-operation F and for ρ-terms τ i (0 i < n), then h(σ) = 1 + max{h(τ i ): i < n}. The depth of a formal equation is the larger of the depths of its two sides. (Thus a linear Mal tsev condition as presented in Theorem is one whose equations are of depth 0 or 1.) We will take d to be the maximum depth of the equations µ j ν j. We then define D to be the set of all terms in this signature, in variables x 0,..., x N 1, and of depth d or less. Clearly D is a finite set.

9 10.11 Taylor classes of varieties 7 Definition of t and s. Now, letting ˆF i denote the term F i (x 0,..., x N 1 ) (for i < m), we define t 0 to be the term ˆF 0 ˆF 1 ˆF m 1, and then we let t = t d 0 (the -product of d copies of t 0 ). This is a V-term with n = N dm variables. Finally we define s = t t, a V-term with n 2 variables. Substitution into t and s. Let X denote an n-tuple of variables selected from {x 0,..., x N 1 }; we may write X as x φ(i) : i < n for an appropriate function φ: n N. Then if u is a term in x 0,..., x n 1, we may use either u(x) or u( x φ(i) : i < n ), or simply u(x φ ) or u(φ) to denote the substituted term u(x φ(0),..., x φ(n 1) ). If 0 q < N 2, then q has a unique representation as q = a + bn with 0 a, b < n. We define η 0 (q) = a and η 1 (q) = b. Then we have θ 0, θ 1 : n 2 n. We therefore may substitute each θ i into s, forming s(θ 0 ) and s(θ 1 ), each a term in n variables. The reader may check that V satisfies s(η 0 ) s(η 1 ) t (see Exercise 5). LEMMA If α is any term in D, there exists a function φ: n N so that V = t( x φ(i) : i < n ) α. Proof. By induction on i (for 1 i d), one may prove that every α in D of depth i has the required form. (In fact the only V-equations used in this argument are the idempotent laws. See Exercise 6.) Now for each α D, we select one φ obeying the conclusion of Lemma 10.4, and designate it φ α. In other words, we now have V = t(x φα ) α for each α D. The variety W. We specify the signature of W to have a single operation S, which is n 2 -ary. The axioms come in three forms: i. If φ and ψ are any maps n 2 n 2, and if V = s(φ) s(ψ), then S(φ) S(ψ) is an axiom of W. Let 0 stand for this set of equations. ii. S(η 0 ) S(η 1 ) is an axiom of W. (This in fact repeats one of the equations from Part i.) Let 1 stand for this equation. iii. Define T to be the W-term S(η 0 ). Then S T T is an axiom of W. Let 2 stand for this equation. It is not hard to see that if A = A; F i i<m V, then A; s A W, and thus that W is interpretable in V. We are interested, however, in establishing the reverse: an interpretation of V in W. For 0 i < m, we define δ i (x 0,..., x N 1 ) to be the substituted W-term T (φ ˆFi ), where the V-terms ˆF i and the substitutions φ α are as defined above. Now for B = B; S W, we define B to be B, δ B i i<m. LEMMA For all B W, we have B V. In other words the W-terms δ i effect an interpretation of V in W.

10 8 Chapter 10 Part of Chapter 10 Proof. We first proceed by induction on i < d to show that for every α D of depth at most i, and any B W, the operations α B and T (φ α ) B are equal. If α is one of the basic terms ˆF i, this holds by definition of δ i. Then we consider the case where α = β(γ 0,..., γ N 1 ), and β and all γ i have depth < i. By induction, we have α B = β B (γ B 0,..., γ B N 1) = T (φ β )(T (φ γ0 ),..., T (φ γn 1 )) = S(φ λ ) for some λ. Now working in the equational theory of V, we calculate s(φ α η 0 ) t(φ α ) α = β(γ 0,..., γ N 1 ) t(φ β )(t(φ γ0 ),..., t(φ γn 1 )) s(φ λ ) for the same λ. Remembering 0 (see Part i of our axioms for W), we see that W satisfies the equation S(φ α η 0 ) S(φ λ ). Now we may say that α B = S(φ λ ) = S(φ α η 0 ) = T (φ α ), as was to be proved. This finishes the inductive argument. To complete the proof of Lemma 10.5, we let µ i ν i be one of the defining equations of V. In V we have that µ i = t(φ µi ) and ν i = t(φ νi ), by Lemma 10.4; hence the equation t(φ µi ) t(φ νi ) holds identically in V, and is equivalent, via t = s(η 0 ) to a linear equation in s, so that T (φ µi ) T (φ νi ) holds identically in W. From our recent inductive argument, we see that B = µ i ν i. Completion of the proof of Theorem We next wish to prove that Mod 0 is active, i.e. contains no trite algebra. Let us assume, for a contradiction, that Q = Q; S Q is a trite algebra satisfying 0 (Part i of the axioms for W). This means that S Q is, say, the q th coordinate projection, for some q < n 2. The equation S(η 0 ) S(η 1 ) implies that q = p+pn for some p < n. From this one easily verifies that Q = S T T, and so in fact Q W. Considering now the algebra Q, it is clear that Q is trite. It follows moreover from Lemma 10.5 that the trite algebra Q lies in V, contrary to our assumption that V is active. This contradiction establishes that Mod 0 is active. Considering the linear equations 0 as they apply to the V-term s, we again see that they rule out the existence of a trite algebra. This can happen only if (10.3) holds for the term s. Thus s is a Taylor term for V, and Theorem 10.2 is proved. One significant conclusion that we may draw from Theorem 10.2 is that the class of all varieties possessing a Taylor term is a Mal tsev class. (Equivalently, to have a Taylor term is a Mal tsev property, as defined near the start of 10.1) To prove this requires knowing that the Taylor class is

11 10.11 Taylor classes of varieties 9 closed under the formation of product varieties, which are defined in the next chapter. Theorem 10.2 is just what is needed to prove closure under product varieties. See Exercise 10 on page 164 where we return to this subject and describe the details. But in fact that proof can be seen as redundant, since we have (since 2016) a proof that the Taylor class is a strong Mal tsev class. This will be a corollary of Theorem 10.7 just below. Exercises Prove the associativity of the operation on terms. 2. Given matrices M and N as defined in Example 2, prove that they satisfy (10.3). Verify that, for this M and N, the equations F (M) F (N) follow from (10.5) and the 3-permutability equations (10.4). 3. Prove Lemma 10.3 and its corollary. 4. Give a detailed argument, based on the Compactness Theorem, for the finiteness conclusion in the proof of Theorem It may help to note that if G is an r-ary operation symbol and 0 g < r, then there is a first-order sentence that says, G projects each r-tuple to its g th coordinate. Too easy??? 5. Prove that V satisfies s(η 0 ) s(η 1 ) t. (As asserted just before Lemma 10.4.) 6. Complete the inductive argument for the proof of Lemma Derive Taylor terms and equations for congruence regularity. More concretely, let V be the variety defined by the equations of Theorem 10.23(iv), say with n = 3. Exhibit a term T in the f n () and the g n (), and exhibit matrices M and N, so that V satisfies T (M) T (N), and so that (10.3) holds. (It may be useful to consider the relationships that are diagrammed in Figure (page 331). Note also that Olšák s Theorem 10.7 gives an immediate answer for M and N, but does not readily yield a concrete expression for T. is this really true? ) **8. A weak near-unanimity term for a variety V is a V-term F (x 0,..., x n 1 ) such that V satisfies the laws F (x,..., x) x, F (y, x,..., x) F (x, y, x,..., x) F (x,..., x, y). Prove that these equations yield no single-equation Taylor condition. A single-equation Taylor condition for this context would comprise a

12 10 Chapter 10 Part of Chapter 10 term T whose only operation symbol is F, together with single-row matrices M, N such that the given equations imply T (M) T (N), and such that (10.3) holds for M and N. (See A. Kazda, Taylor term does not imply any nontrivial linear one-equality Maltsev condition, **9. (Miroslav Olšák, private communication.) Any Taylor condition with n variables implies a Taylor condition with two equations and n 2 variables. ( Two equations means two equations plus the equation of idempotence.) Hint? Two difficult exercises. I could see dropping them, or adding hints. Olšák s Theorem For M and N two m n matrices of variables, satisfying (10.3), let us define T M,N to be the class of all varieties V with an (M, N)-Taylor term, i.e. those with an n-ary term T satisfying T (M) T (N). As remarked above, T M,N is the strong Mal tsev class defined by T (M) T (N). Now the class of all Taylor varieties, T = T M,N (the union over all appropriate M and N), is ostensibly not a Mal tsev class. Nevertheless, it turns out that T is a strong Mal tsev class. According to a recent and surprising theorem of M. Olšák there are 2 12 matrices M 0, N 0 such that T = T M0,N 0. Equivalently, T M,N T M0,N 0 for all appropriate M and N. Another way to say it is that if a variety V has any Taylor term, then V has an (M 0, N 0 )- Taylor term. Thus T (M 0 ) T (N 0 ) may be said to define the weakest Taylor condition. We begin our discussion of Olšák s Theorem with a description of M 0 and N 0. Let K be the 4 12 matrix whose columns are all the four-tuples a, b, c, d {x, y} 4 with a b or c d, organized in some reasonable list from left to right (there are twelve such columns). Let r 0, r 1, r 2, r 3 be the first, second, third and fourth rows of K, respectively. Olšák s equations are then S(r 0 ) S(r 1 ); S(r 2 ) S(r 3 ). Organizing the columns alphabetically, the equations become S(x, x, x, x, x, x, y, y, y, y, y, y) S(x, x, y, y, y, y, y, y, x, x, x, x) (10.9) S(x, y, x, x, y, y, x, y, x, x, y, y) S(y, x, x, y, x, y, y, x, x, y, x, y). (10.10)

13 10.11 Taylor classes of varieties 11 Or, to describe the matrices directly, we have ( ) x x x x x x y y y y y y M 0 = ; x y x x y y x y x x y y ( x x y y y y y y x x x x N 0 = y x x y x y y x x y x y THEOREM (M. Olšák, 2016) Every Taylor condition implies the (M 0, N 0 )-Taylor condition. Proof. Let us be given (M, N, F )-Taylor equations F (M) F (N) as defined in Definition 10.1, where M and N are n n matrices with x s on the diagonal of M and y s on the diagonal of N. We shall prove the existence of an F -term S satisfying Equations ( ). In fact, we shall prove that (10.9) is a consequence of the idempotence of F, and (10.10) is a consequence of F (M) F (N). We begin with the construction of some algebras in the signature of a single n-ary operation F. Let A be the free idempotent algebra of this signature freely generated by {x A, y A }. Let B be the free algebra of the same signature, freely generated by {x B, y B } in the variety axiomatized by the equations F (M) F (N). Let C = A B. We define a subalgebra of C. Take Q = { a, b, c, d {x A, y A } 2 {x B, y B } 2 : a b or c d}, and define P as the subalgebra of A 2 B 2 generated by Q. Our goal is to show that P contains a 4-tuple a, a, b, b (what Olšák calls a double loop ). This will mean there is an F -term S such that A = S(x, x, x, x, x, x, y, y, y, y, y, y) S(x, x, y, y, y, y, y, y, x, x, x, x) B = S(x, y, x, x, y, y, x, y, x, x, y, y) S(y, x, x, y, x, y, y, x, x, y, x, y). Theorem 10.7 is an immediate corollary of this fact. The proof now divides into two parts. In the first part, we construct a special subalgebra of A 2 ; see Lemma In the second part, we shall prove that the graph A, R of Lemma 10.8 has a loop: a, a R for some a (see Lemma 10.10). LEMMA There is a subuniverse R of A 2 satisfying i. R is symmetric: a, b R implies b, a R. ii. For all a, b R there exists c B so that a, b, c, c P. iii. The graph A, R has an odd cycle. iv. R T A 2 (R absorbs A 2 with respect to T ). This means that whenever a 0,..., a n 1 A 2 and a i R for all but at most one i, then T (a 0,..., a n 1 ) R. ).

14 12 Chapter 10 Part of Chapter 10 Proof. To construct R to satisfy Lemma 10.8, we recursively define a sequence of elements σ j (for j ω) as follows. To begin, we note that there is a unique automorphism of B, b b, that satisfies x B y B and y B x B. We take σ 0 = x B. Now suppose that j 0 and j = mn + i with 0 i < n. We define σ j+1 as follows. We write our i th Taylor equation (encoded in the i th rows of M and N) as L i (x, y) = T (row i of M) = T (row i of N) = R i (x, y). I deleted McKenzie s r i, t i since they never appeared again in the proof. We now put σ j+1 = L i (σ j, σ j ). Now we put R j = { a, b A 2 : a, b, σ j, σ j P }; R = j ω R j. Note that R 0 contains the pair x A, y A. In due course we will see that R j R j+1 for all j, and that R has the properties listed in Lemma We now consider three automorphisms of A 2 B 2. First, we have a, b, c, d a, b, c, d (with c and d as defined above). Second is a, b, c, d b, a, c, d, and third is a, b, c, d a, b, d, c. LEMMA i. The algebra P is invariant under each of the three automorphisms of A 2 B 2 defined above. ii. For all a, b A, there exists m such that for all k m we have a, b, σ k, σ k P and a, b, σ k, σ k P. Proof. Statement (i) follows from the observation that the generating set Q is invariant under each of the three automorphisms. For (ii) we will use the -product introduced in the proof of Theorem For m 1 we will consider the term T m, which is the m-fold - product of T with itself. Its arity is n m. In particular, we write T m (X) for the set of all values T m (u 0,..., u nm 1) where u X mn. Here we shall have X contained in either A, A 2, B, B 2 or C = A 2 B 2, with the term to be evaluated in the appropriate algebra. It is easily proved by induction on m that and in fact σ m+1, σ m+1 T m ({x B, y B }) for all m ω, σ m+1, σ m+1 T m ({ x B, y B, y B, x B }) for all m ω. Moreover, due to the idempotence of A, the set {x A, y A } 2 generates A 2 and every a, b A 2 belongs to T m ({ x A, y A } 2 ) for m sufficiently large. For m that large, we have that { a, b, σ m+1, σ m+1, a, b, σ m+1, σ m+1 } T m ({x A, y A } 2 { x B, y B, y B, x B } P.

15 10.11 Taylor classes of varieties 13 This proves (ii) and completes the proof of Lemma 10.9 Returning to the proof of Lemma 10.8, we note that each R j is symmetric, by Part (i) of Lemma Therefore R, the union of the R j s is itself symmetric (and Part (i) is proved). For Part (ii) of Lemma 10.8, we refer only to the definition of R j, where to have a, b R j we required that a, b, σ j, σ j belong to P. Now Part (ii) holds because R is the union of the sets R j. As promised, we show that R j R j+1 for each j. Let us say that j = mn + i with m ω and 0 i < n. For a, b R j, we have a, b, σ j, σ j P. By Lemma 10.9(i) we have a, b, σ j, σ j P. Now a, b, σ j+1, σ j+1 = L i ( a, b, σ j, σ j, a, b, σ j, σ j ) P. Thus a, b R j+1. We now turn our attention to Part (iv) of Lemma To show that R absorbs A 2 with respect to T, we will let 0 i < n and assume that a0, b 0,..., a n 1, b n 1 = a, b (A 2 ) n, with a u, b u R for u i. We need to show that T (a), T (b) R. By Part (ii) of Lemma 10.9 and the fact that R j is an increasing sequence of sets, we can choose m so that m = pn + i and { au, b u,σ m, σ m } { au, b u } { σ m, σ m, σ m, σ m } u i = Γ P. It should be obvious that 0 u<n T (a), T (b), σ m+1, σ m+1 = T (a), T (b), L i (σ m, σ m ), R i (σ m, σ m ) T m (Γ) P. This is because at position i, expressed as instances of T, L i (x, y) has an x and R i (x, y) has a y. Thus, indeed, T ( a 0, b 0,..., a n 1, b n 1 ) P. We now turn our attention to Part (iii) of Lemma We will show that the graph A, R has an odd cycle (actually a (2n + 1)-cycle). We first observe that x A, y A, σ 0, σ 0 Q P, and hence x A, y A R. To simplify notation, we shall finish the proof of Lemma 10.8 writing x for x A and y for y A. We have T (x,..., x), T (x, y,..., y) R, since x, y R and R absorbs x, x. Thus x = T (x,..., x) R T (x, y,..., y) R T (y, x,... x). To continue another two steps, with similar justification, we have T (y, x,..., x) R T (x, x, y,..., y) R T (y, y, x,... x). After 2n + 1 steps we have x = T (x,..., x) R 2n T (y,..., y) = y R x, an R-cycle of odd length. This ends the proof of Lemma 10.8.

16 14 Chapter 10 Part of Chapter 10 We now enter the second (and final) part of the proof of Theorem We will prove that R contains a loop, in other words, that a R a for some a. According to Part (ii) of Lemma 10.8, this means that there exists c C such that a, a, c, c P. And according to the remarks just before Lemma 10.8, the existence of such a double loop will guarantee the existence of the desired Olšák-term S, thereby completing the proof of Theorem We begin with some remarks, and a lemma, on graph theory. Suppose g : K n K, and L K. We say L absorbs K with respect to g, and write L g K, to mean the following: if k j K (for 0 < j < n), and if k j L for all but at most one value of j, then g(k 0,..., k n 1 ) L. I couldn t find a definition of g in the McKenzie writeup, although what I have written here extrapolates from the definition for ordered pairs that appears in Lemma I think it will work for what follows, but that needs to be confirmed as we go along. I am somewhat guessing how to put the symbol into words. Tried finding it in the literature no luck. WT Let X, E be a symmetric graph. For x X, write N(x) = {y X : x, y E} (the set of neighbors of x). Write N( X, E ) for the set of x X such that N(x) (the domain of E). Suppose that g : X k X for some k 1. We say that E semi-absorbs with respect to g iff for all x X, N(x) g N( X, E ). Notice that for the R of Lemma 10.8, R semi-absorbs with respect to T. (This is easily seen to be implied by R T A 2, and the idempotence of T from Lemma 10.8.) Thus the next lemma completes the proof of our main result, Theorem LEMMA Suppose that A, R is a symmetric graph with an odd cycle and that g : A n A for some n 1. Suppose further that R is a subuniverse of A, g 2 and R semi-absorbs with respect to g. Then R has a loop: a R a for some a A. Proof. Note that we are not assuming idempotence for g in this lemma. Notice also that A is non-empty (because of the cycle), and by replacing A with N(A) (which is a g-subuniverse), we may assume that N(A) = A. For n 1 and l an odd number 1, we let P (n, l) stand for the validity of the Lemma in the case of n-ary g and the existence of an odd cycle of size l or smaller. The proof will be a double induction on n and l, organized as follows. We begin with an easy direct argument for P (1, 1). Then in proving P (n, l) we allow ourselves to assume the truth of P (n, l ) for any smaller l, and the truth of (n, l ) for any l and any smaller n. (This is induction over the well-ordered set ω ω.) If n = 1, then semi-absorption amounts to this: If x, y A, then g(y) N(x). Taking x = g(y), we have a loop g(y) R g(y). Thus we have a basis for our induction. Our inductive proof now divides into three cases: (a) l = 1, (b) l > 3, and (c) l = 3.

17 10.11 Taylor classes of varieties 15 In Case (a), we have a cycle of length 1, i.e. a loop, and the lemma holds. For the next case, l > 3, we proceed as follows. We replace R by R = R 3. Note that R R because R is symmetric: a, b R yields a path a R b R a R b, giving a, b R. Now let a 0 R a 1 R R a l 1 R a l = a 0 be a cycle in R of length l. Then a 0 R a 3 R a 4 R R a l = a 0 is an R -cycle of length l 2. Moreover, it is easy to see that g respects R and R semi-absorbs with respect to g. By induction, it follows that R has a loop: a R a. This means that there exist b, c such that a R b R c R a, i.e. that R has a 3-cycle. Now by induction R has a loop. We now come to Case (c), the proof of P (n, l) when l = 3. So A, R has a triangle a R b R c R a. We call a, b, c a weak 3-clique. ( Weak because it is possible that some two of a, b, c are equal of course, if that is the case we have a loop and we are done.) In general, a weak k-clique in A, R is a sequence a 0,..., a k 1 with a i R a j whenever 0 i < j < k. We now have a weak (0 + 3)-clique. Claim. If A, R has a weak (i + 3)-clique, then it has a weak (i )- clique. To prove the claim, we begin by writing the given (i + 3)-clique as a 0, a 1,..., a i+2, where a j R a k when 0 j < k < i + 3. We will write this also as a 0,..., a i 1, a, b, c. Now let A = N(a 0 ) N(a 1 ) N(a i 1 ) N(a). Note that A is a subuniverse of A, g due to semi-absorption, and A {b, c} and b R c. Put R = (R (A ) 2 ) 3 (3-fold relational product), and g a (x 0,, x n 2 ) = g(x 0,, x n 2, a) for x 0,..., x n 2 A. In subsequent paragraphs, we will show that A, R, g a satisfy the hypotheses of P (n 1, l ) for some l. Then by induction, we know that R has a loop, i.e. there are a, b, c A such that a R b R c R a. Thus a 0, a 1,..., a i 1, a, a, b, c is a weak (i+1+3)-clique in A, R. This finishes the Claim, modulo the hypotheses of P (n 1, l). First, we need to show that R has an odd cycle that lies completely in A which will be a cycle for R, since R restricted to A is a subset of R. Indeed we have g(b,..., b) R g(a, c,..., c) R g(c, b,... ) R g(b, a, c,..., c) R g(c, c, b,..., b) R R g(c,..., c) R (b,..., b). This is a cycle of length 2n + 1, and each of its elements is in A, by semi-absorption.

18 16 Chapter 10 Part of Chapter 10 That A is closed under the action of g a follows from semi-absorption with respect to g. That R is semi-absorbed??? semi-absorbs??? with respect to g a is seen as follows. Suppose p, a 1,... a n 2, u are elements of A, and that a 1,..., a n 2 are R -neighbors of p. We can exhibit these neighbor relations as a i R b i R c i R p (for 1 i < n 1), with b i, c i A. We also have u R a R b R c and a R b R a R p these follow from our choice of a, b, c and from the definition of A. Thus g(a, u, a) R g(b, a, b) R g(c, b, a) R p, by semi-absorption, since c 1,..., c n 2 are all neighbors of p. All four of the displayed elements belong to A, by semi-absorption. Thus the displayed relations show that g a (a 1,..., a n 2, u) is an R -neighbor of p. Thus R semi-absorbs with respect to g a. That R respects g a is shown like this: Suppose that a i, d i R for 0 i < n 1. Thus we have b i, c i A with a i R b i R c i R d i. We also have a R b R c R a. Now consider g a (a) = g(a, a) R g(b, b) R g(c, c) R g(d, a) = g a (d). All elements named in the displayed relations belong to A by semiabsorption. Thus g a (a), g a (d) R as desired. This concludes the proof of the Claim. Continuing with the proof of Lemma 10.10, we observe from the Claim (and induction) that A, R has a weak n-clique: a 0,..., a n 1 with a i R a j for 0 i < j < n. Put p = g(a 0,..., a n 1 ). For each i, it follows by semi-absorption that p is a neighbor of a i. But then p = g(a 0,..., a n 1 ) is a neighbor of p (a trivial consequence of semi-absorption). So we have p R p, the desired loop, and the proof of Lemma is complete. As remarked above, this loop in R yields a double loop in P, which in turn completes the proof of Olšák s Theorem COROLLARY The class T of all varieties that satisfy some Taylor condition (see page 10) is a strong Mal tsev class. Proof. T is the class of all varieties in which one can interpret the equations ( ) on page 10, along with idempotence. Exercises Prove that, as claimed early in the proof of Theorem 10.7, the discovery of a double loop leads to the existence of an idempotent S satisfying ( ). In particular, confirm that the pattern of x s and y s is exactly as determined by the twelve elements of Q. *2. Exhibit an Oršák term for congruence k-permutability, for arithmeticity, and for xxxxxxxx. ( Check feasibility, and exact scope, of this exercise..)

19 C H A P T E R E L E V E N Abstract Clone Theory Many phenomena encountered by the student of varieties are most naturally viewed and studied in the framework of clone theory. Some aspects of varieties demand for their full appreciation a good familiarity with clones; for example, the equivalence of varieties ( 4.12 of Volume 1), many of the properties that were used to classify varieties in Chapter 10, e.g. Mal tsev properties, and all of the known constructions that combine two or more varieties to produce a new variety. In Chapter 9 we continued the study of clones that we began in Volume 1, again in the down to earth setting of clones of operations. The dominant themes of that chapter were the clone of all operations on a finite set k, the structure of its subclones, and the polarity between operations and relations on k. In this chapter, we shall be concerned with clones in their full abstract generality, and with operations such as direct product, free product and tensor product that can be applied to clones, and consequently, to varieties. Except in the first section, we emphasize the close parallel that exists between the theory of clones and the general theory of varieties. For example, there is a natural one-one correspondence χ between isomorphism classes of clones and equivalence classes of varieties (in the sense of 4.12). This correspondence is presented in Theorems?? and of 11.2 below. An interpretation of one variety V in another variety W corresponds exactly to a clone homomorphism h : χ(v) χ(w). We will study the quasi-order on the class of all varieties that is defined by V 1 V 2 iff V 1 is interpretable in V 2. (Under χ, there is a corresponding quasi-order on clones, defined by C 1 C 2 iff there exists a homomorphism C 1 C 2.) The Mal tsev properties introduced in 4.12 (and discussed at greater length in Chapter 10) are most naturally defined as certain kinds of filters over this quasi-order. Co-authors: all introductory paragraphs beyond this point subject 17

20 18 Chapter 11 Abstract Clone Theory to eventual modification, removal or displacement. It would be wrong to regard clones as just a high level abstraction with potential for organizing our knowledge of varieties. Rather, clones and varieties are like the two sides of a coin. The concept of a clone is actually in several ways more concrete and readily accessible to the intuition than is the concept of variety. On the other hand, the fact that clones come with varieties attached to them is very useful in their study. For example, the coproduct of two clones is easily seen to exist, but is difficult to visualize, while the variety correlated with the coproduct is easily defined and manipulated. For products of varieties, the tables are turned, and the clone-theoretic viewpoint has the conceptual advantage. It is natural and instructive to develop clone theory and the theory of varieties in parallel, switching from one perspective to the other whenever it seems convenient. (We did not do this from the start of these volumes, because we felt that one powerful abstraction at a time would be quite enough for the student beginning variety theory.) Clone theory, as presented in this chapter, is virtually a theory of varieties of algebras. In our view, it complements, but will never entirely replace the dominant model-theoretic approach to varieties that we have followed throughout the earlier parts of this work. We should point out that some mathematicians have believed otherwise, and indeed the pioneers of this theory, especially the school of F. William Lawvere, studied clones under the name of algebraic theories for their own intrinsic interest, quite apart from any possible application to varieties. We hope that our readers will be persuaded by the end of this chapter that the vast algebraic landscape can be surveyed from at least three vantage points, namely with the emphasis on algebras, or varieties, or clones. The ways of thinking proper to each of these perspectives should be part of one s mathematical toolkit. Although there are apparently a large number of details in the first section, no individual idea will be particularly difficult for a student who has mastered, say, 3.6, and of Volume 1. The first section contains all the ideas of pure clone theory; in our applications in later sections, we will be able to speak of clones with no need to supply every detail. For the general idea of such notions as homomorphisms, subclones, and the subclone generated by a subset, the reader can in large part rely on his intuition and on his experience from Volume 1, while referring to the precise definitions whenever necessary. Nevertheless, as a minimum requirement for leaving the first section and proceeding to the rest of the chapter, the reader should We also recommend Examples.

21 11.1 Abstract Clones and their Homomorphisms Abstract Clones and their Homomorphisms Version of 13/11/2018(format: day/month/year). Clones as we defined them at the start of Chapter 4 (see page 143 of Volume 1) serve as a prototype for the corresponding abstract notion. The definition introduced there (and followed so far in this Volume, especially in Chapter 9) says that a clone is a collection of operations, defined on a nonvoid set A, that contains the projection operations and is closed under all compositions. To underline the distinction between clones of this sort and the abstract clones that we are about to define, we will in this chapter refer to the former as concrete clones or clones of operations. The modern concept of a group evolved from the earlier notion of a group of permutations, and the abstract and concrete notions are linked by the Cayley representation of 3.5; there is a similar relationship between abstract semigroups and semigroups of maps (i.e. subsemigroups of End A). In like manner, we may form abstract mathematical objects that contain the essential features of clones of operations. In the manner of Cayley s Theorem, each clone, in this abstract sense, may be represented as isomorphic to a clone of operations on a set. (See the corollary to Theorem on page 46.) We believe that there are two valuable ways to formulate abstractly the composition of operations; they take into account in two different ways the fact that composition is not everywhere defined. One formulation proceeds from the following simple observation. A clone U of operations on a set A determines, and is determined by, the subcategory of SETS whose objects are the finite direct powers A m (m 1) and whose morphisms are the functions f : A n A m (m, n 1) such that p m i f U for every projection operation p m i (see pages of Volume 1, and also Exercise 15 on page 140 there). In this formulation, there is only a single operation (the binary product operation in a category), and that operation is only partially defined, as is standard for categories. The other formulation is more elementary, but also more cumbersome, since it requires explicit mention of composition operations for all arities, and requires us to explicitly mention a domain for each composition. This formulation of composition causes abstract clones to take the shape of multi-sorted or heterogeneous algebras. A multi-sorted algebra has an indexed family U i : i I of universes, and each of its operations is a function f : D E, where D is a finite product of some of the universes U i, and E is one of the universes U i. (See Exercise 11.8(11.8) for more information about multi-sorted algebras.) It is apparent from the opening remarks of Chapter 4 (pages of Volume 1) that a clone U of operations on a set A naturally fits this pattern, if for i = 1, 2,... we take U i to be the collection of i-ary operations in U, and if for all k, n 1 we take one of our clone operations to be the operation Fn k of constructing an n-ary operation h from one k-ary operation g and k n-ary operations

22 20 Chapter 11 Abstract Clone Theory f 0,..., f k 1 by the rule h(x) = g(f 0 (x),, f k 1 (x)). (11.1) In other words, the domain D of Fn k is U k (U n ) k, its codomain E is U n, and Fn k : D E is defined via Fn k (g, f 0,, f k 1 ) = h. As in Volume 1, we may also denote the operation h defined in this way as g(f 0,, f k 1 ). Because each of these formulations is useful, we will give two definitions, which together incorporate both the categorical and the settheoretic points of view. Our first definition recapitulates material on page 136 of Volume 1. DEFINITION A (full abstract) clone consists of a small category C = C,, with designated C-objects O i (i = 1, 2, ) and designated C-morphisms p j i hom C(O j, O 1 ) for 1 j < ω and 0 i < j, satisfying i. Each O j is the product of the object O 1 (taken j times) with associated projection morphisms p j 0,, pj j 1. ii. The correspondence j O j integers and the objects of C. is a bijection between the positive A category is small if, as a class of morphisms, it is a set and not a proper class. Many of our constructions, such as forming the clone of a variety, or a clone of operations on a given set A, naturally give rise to a category that is small. Thus smallness is not a serious limitation of our methods and results. Generally we let a single bold letter, such as C here, denote both a clone and its underlying small category. We now present the alternative axiomatization of abstract clone theory that was promised on page 317 of Volume 1. DEFINITION A (basic abstract) clone is a disjoint family of sets U 1, U 2,... together with, for each k, n = 1, 2, a function F k n : U k (U n ) k U n and for each j = 1, 2,, elements p j 0, pj 1, pj j 1 U j, such that the following equations hold for all relevant i, j, k, s, n, and whenever all F - values are defined: F s n(x, F j n(y 0, z 0,, z j 1 ),,F j n(y s 1, z 0,, z j 1 )) F j n(f s j (x, y 0,,y s 1 ), z 0,, z j 1 ) (11.2) F j k (pj i, x 0,, x j 1 ) x i (11.3) F j j (y, pj 0,, pj j 1 ) y. (11.4) (One may check that, given the indicated domains of the various F k n, the left-hand side of (11.2) is defined for given values x, y 0, y 1,, z 0, z 1,,

23 11.1 Abstract Clones and their Homomorphisms 21 if and only if its right-hand side is defined for those values.) The morphisms p j i are called projections and the functions F k n are called composition operations. In this chapter we will generally omit the word abstract. The modifier full or basic can often be inferred from context, and in that case may be omitted. In fact we may easily pass between the two sorts of abstract clone, as described in the next two theorems. For this reason we may in many cases use the two notions interchangeably. THEOREM Let C = C,, O i, p j i be a full clone. Then U i, F k n, p j i is a basic clone, where U i = hom C (O i, O 1 ) and where F k n (k, n = 1, 2, ) is defined as follows. F k n (g, f 0,, f k 1 ) = g f, (11.5) where f is the unique member of hom C (O n, O k ) such that p k j f = f j for 0 j < k. Proof. Unique existence of f follows from the fact that the projections p k 0, p k 1,, p k k 1 express O k as the k-fold product of O 1 with itself; thus F k n is well defined. We need only verify Equations 11.2, 11.3, and 11.4 to complete the proof. We will leave the verifications of 11.3 and 11.4 to the reader, while concentrating on So suppose that z m (0 m < j), y i (0 i < s) and x are arbitrary elements (in the appropriate sets U i for Equation 11.2). Define z hom C (O n, O j ) and y hom C (O j, O s ) to be the unique morphisms such that p j m z = z m (0 m < j) and p s i y = y i (0 i < s). In other words, we have O n z O j y O s x O 1. Now Equation 11.2 is easily translated into (x y) z = x (y z), which follows from the associative law for categories. DEFINITION In the situation of Theorem 11.3, we say that U i, Fn k, p j i is the basic part of C, or that C is an expansion of U i, Fn k, p j i (to a full clone). THEOREM Every basic clone has an expansion to a full clone, which is unique in the following sense. If C and D each expand U i, F k n, p j i, then there is a category isomorphism F : C D such that, for each j, F is the identity on U j = hom C (O j, O 1 ) = hom D (O j, O 1 ). Proof. Suppose that operations F k n on sets U i satisfy equations 11.2, 11.3

24 22 Chapter 11 Abstract Clone Theory and We may define a category C by taking the universe to be 1 C = 1 n<ω 1 k<ω (U n ) k. (11.6) For x = x 0,, x m 1 (U s ) m and y = y 0,, y t 1 (U j ) t, we stipulate that the product x y is undefined if t s, and equal, if t = s, to w 0,, w m 1 where, for 0 i < s, w i = F s j (x i, y 0,, y s 1 ). (11.7) To see that the partial algebra C = C, formed in this way is a category, we need to check Axioms (i)-(v) of 3.6 (page 133 of Volume 1). The first of these axioms says that if x y and (x y) z are defined, then so are y z and x (y z), and x (y z) = (x y) z. So, let us suppose that x and y are as specified above, and that z = z 0,, z r 1 (U k ) r. The assumptions that x y and (x y) z are defined tell us that t = s and r = j. From these two equations, it readily follows that y z and x (y z) are defined as well. Now the desired equation x (y z) = (x y) z can be regarded as equivalent to the system of equations x i (y z) = (x i y) z (0 i < m), (11.8) where each x i is regarded as a member of (U s ) 1, and hence as an element of C. Now, the reader may easily check that if Equations 11.8 are expanded according to the definition of the category operation (i.e. according to Equation 11.7), then one obtains precisely Equation 11.2 with x replaced by x i which we have assumed for the operations Fn k. Therefore the first category axiom is established. The proofs of the other category axioms are similar, and may be left to the reader. (For instance, the units of C have the form p j 0,, pj j 1.) Considering now the uniqueness of C up to isomorphism, 2 we assume that D = D,, Q i, p j i is another expansion of U i, Fn k, p j i. We define φ : C D as follows. If x is any element of C, we have x hom C (O k, O j ) for some k and j. Then, for each i < j, p j i x lies in hom C(O k, O 1 ) = hom D (Q k, Q 1 ). Since Q k is the product of k copies of Q 1 with projection maps p j i, there exists a unique x hom D (Q k, Q j ) such that p j i x = p j i x for 0 i < j. We then define φ(x) to be x. To examine the value of F (x) for x U k = hom C (O k, O 1 ) = hom D (O k, O 1 ), we consider the foregoing with j = 1. For i = 0 we have p 1 0 x = p 1 0 x in U 1 U k. Now p 1 0 is a bijection, and hence x = x. in other words F acts identically on U k. It remains to show that φ is a functor. This is a straightforward application of Equation 11.5, which we may leave to the reader. Since an inverse to φ 1 We are taking (U 1 ) 1 to be equal to U 1. 2 This last part of the proof is essentially a solution to Exercise 14 on page 140 of Volume 1.