THE AMALGAMATION CLASS OF A DISCRIMINATOR VARIETY IS FINITELY AXIOMATIZABLE. Clifford Bergman

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1 THE AMALGAMATION CLASS OF A DISCRIMINATOR VARIETY IS FINITELY AXIOMATIZABLE Clifford Bergman Discriminator arieties hae been extensiely studied since their introduction by Pixley in Among their attributes, discriminator arieties exhibit a strong relationship between the quantifier-free formulas and certain terms in the language of the ariety. This paper exploits that relationship in order to proe that some important classes of algebras, generally defined using algebraic properties, can be described by a finite set of first-order sentences. If K is a class of algebras, define Ps(K) to be the class of all algebras isomorphic to a subdirect product of members of K. AnalgebraAof K is an amalgamation base of K if, for eery B 0, B 1 Kand α 0,α 1 embedding A into B 0 and B 1 respectiely, there is an algebra C of K and embeddings β 0 of B 0 into C and β 1 of B 1 into C such that β 1 α 1 = β 0 α 0. The collection of all amalgamation bases is called the amalgamation class, Amal(K). Let V be a finitely generated discriminator ariety of finite type. In this paper, the following classes will be shown to be finitely axiomatizable: (1) Ps(S), where S is a set of simple algebras of V; (2) Amal(V). These results are taken from the author s doctoral thesis written under Ralph McKenzie. The author is greatly indebted to him for his guidance and suggestion of this problem. Notation. For a class K of algebras, K si denotes the class of subdirectly irreducible algebras of K. Terms in the language of K will be denoted by lower case Greek letters. If α is a term and A Kthen α A denotes the term function of A corresponding to α. For conenience, the sequence of letters x 0,x 1,...,x n 1 will often be abbreiated x. The lattice of congruences of an algebra A is denoted by Con(A) and has smallest element and largest element. IfBis a subalgebra of A and Θ Con A, then Θ B denotes the congruence Θ B 2 of B. For definitions and basic facts not explained here, the reader can consult 2] or 3]. Definition 1. AarietyVis a discriminator ariety if there is a term σ(x, y, z) in the language of V (called the discriminator term) such that an algebra A V is subdirectly irreducible or triial if and only if A ( σ(x, x, z) z x y σ(x, y, z) x ). Typeset by AMS-TEX 1

2 In 6] Pixley showed that a finitely generated ariety is a discriminator ariety if and only if it is arithmetical, and eery subalgebra of a subdirectly irreducible algebra is either simple or triial. Suppose (A i : i I) is a family of algebras and D is an ultrafilter on the set I. Then D induces a congruence (also denoted D) on (A i :i I)bya b (mod D) if and only if { i I : a i = b i } D. To build a set of sentences describing Ps(S), we first need sentences to insure that the discriminator term has the desired properties. These were discoered by R. McKenzie 5]. Theorem 2. Let L be a first-order language with no non-logical relation symbols and with function symbols { f i : i<k},ka cardinal. Suppose σ(x, y, z) is a term of L and let Σ be the set consisting of the following identities (e0) σ(x, x, y) y (e1) σ(x, y, x) x (e2) σ(x, y, y) x (e3) σ(x, σ(x, y, z), y) y (e4) i σ(x, y, f i ( 0, 1,..., ni 1)) σ ( x, y, f i σ(x, y, 0 ),...,σ(x, y, ni 1)] ) for i<k,wheref i is n i -ary. (1) The ariety V determined by the equations Σ is a discriminator ariety with discriminator term σ. (2) Eery finite algebra of V is a direct product of simple algebras. (3) For eery A V and a, b A, the binary relation Θ(a, b) := { (x, y) A 2 : σ A (a, b, x) =σ A (a, b, y) } is the smallest congruence of A containing (a, b). (4) For eery quantifier free formula φ of L, there are terms α and β of L such that for eery B V si and b 0,...,b m 1 B, B φ( b) ( y)α( b, y) β( b, y). Corollary 3. Let V be a discriminator ariety with discriminator term σ. Then the equations of Σ hold in V. Letφ,αand β be as in Theorem 2(4). For any A V and a 0,...,a m 1 A, the following are equialent: (1) there exists a coatom Ψ of Con A such that A/Ψ φ(a 0 /Ψ,...a m 1 /Ψ); (2) A ( y) α(a 0,...,a m 1,y) β(a 0,...,a m 1,y). Proof. Let A V and suppose (i) holds. Take B = A/Ψ andb i =a i /Ψfor i<m. Then B φ(b 0,...,b m 1 ) implies by Theorem 2 that there is z B with α B (b 0,...,b m 1,z) β B (b 0,...,b m 1,z). Choose y A with y/ψ =z. Since α A ( a, y)/ψ =α B ( b, z) β B ( b, z) =α B ( a, y)/ψ we conclude (ii). 2

3 Conersely, if α and β disagree for some y A, then they are separated by a completely meet-irreducible congruence Ψ. By semi-simplicity, B = A/Ψ issimple, and we reerse the implication aboe to derie (i). For the remainder of this paper, suppose V is a finitely generated discriminator ariety of finite type, that is the language of V has only finitely many basic operation and constant symbols. If B is a finite structure of this language, then there is a quantifier-free formula Dg B (x 0,x 1,...,x n 1 ) (the diagram of B ) such that, for eery structure A and a 0,a 1,...,a n 1 A, A Dg B ( a) ifandonly if {a 0,a 1,...,a n 1 } is the unierse of a subalgebra of A isomorphic to B. Setting φ equal to Dg B in the Corollary, we obtain terms α and β and elements a 0,a 1,...,a n 1,b of A with α A ( a, b) β A ( a, b) if and only if, for some coatom Ψ of Con A, A/Ψ contains a copy of B as a subalgebra. What is more surprising, we can strengthen this inequation in such a way that A/Ψ will be isomorphic to B. This idea is incorporated into the following theorem. Theorem 4. Let V be a finitely generated discriminator ariety of finite type. Let S V si.thenps(s) is a finitely axiomatizable class. Proof. Since V is finitely generated, we may assume that S is a finite set of finite algebras, S = {L 0,...,L m 1 }. Since the language is of finite type, the set Σ of Theorem 2 is finite. Informally, we need to add to Σ a sentence saying that for any pair of distinct elements c and d, there is a completely meet-irreducible congruence Ψ separating c and d so that the quotient algebra modulo Ψ is isomorphic to one of the L j s. Fix j<m.letl=l j and r =card(l). By Corollary 3, there are terms α and β such that for eery A Σanda 0,a 1,...,a r 1,c,d A A ( y)α( a, y, c, d) β( a, y, c, d) if and only if there is a coatom Ψ of Con A such that A/Ψ c/ψ d/ψ Dg L (a 0 /Ψ,...,a r 1 /Ψ). Now define the formula Sep L (u, ) tobe ( x 0,x 1,...,x r 1 )( y)( z) α( x, y, u, ) β( x, y, u, ) ( σ(xi,z,α( x, y, u, )) σ(x i,z,β( x, y, u, )) )]. i<r The key claim is that for any A Σandc, d A, A Sep L (c, d) if and only if there is a coatom Ψ of Con A such that c/ψ d/ψ anda/ψ =L. Once this is established, the members of Σ together with the sentence will axiomatize Ps(S). ( u)( ) ( u j<m Sep Lj (u, ) ) 3

4 Suppose first that for c, d A, thereisψ Con A such that c/ψ d/ψ and A/Ψ =L. Then A/ΨsatisfiesDg L for some elements g 0,g 1,...,g r 1. Choosing a i in A to represent g i modulo Ψ, there is a b A such that α A ( a, b, c, d) β A ( a, b, c, d). Now let e be any element of A. SinceA/Ψ ={g 0,g 1,...,g r 1 },there is i<rsuch that e a i (mod Ψ). For this i we hae σ A (a i,e,α A ( a, b, c, d)) σ A (a i,e,β A ( a, b, c, d)) since they are incongruent modulo Ψ. Thus A satisfies Sep L (c, d). Conersely, let A Sep L (c, d). Let a 0,a 1,...,a r 1,b be elements that witness the existential quantifiers. Denote the elements α A ( a, b, c, d) andβ A ( a, b, c, d) by α and β respectiely. Write A as a subdirect product of subdirectly irreducible algebras, A (A t : t T ), and let Θ t be the kernel of the projection A A t. Set U = {t T : α β (mod Θ t ) } and for eery x A, V x = { t T :forsomei<r,a i x (mod Θ t ) }. Claim. There is an ultrafilter D on T such that U D and for eery x A, V x D. Once the claim is established the Theorem will follow easily. For, take Ψ = D A. Ψ is a coatom of Con A since A/Ψ can be embedded in the ultraproduct ( A t )/D which is simple (V is finitely generated). Since U D, α β (mod Ψ), hence by the construction of α and β, c/ψ d/ψ and{a 0 /Ψ,...,a r 1 /Ψ} forms a subalgebra of A/Ψ isomorphic to L. Butforeeryx A,V x Dimplies that x a i (mod Ψ) some i<r. Therefore L = {a 0 /Ψ,...,a r 1 /Ψ}=A/Ψ and the Theorem follows. To erify the claim, it suffices to show that the family {U} {V x :x A} has the finite intersection property. For this choose x 0,x 1,...,x k 1 from A for some natural number k and let E be the subalgebra of A generated by all the elements a 0,a 1,...,a r 1,b,x 0,x 1,...,x k 1,αand β. This is a finite set so, since V is finitely generated, E is finite. Therefore, by Theorem 2(2), E is a direct product of simple algebras, in fact E = (E t : t T 0 )wheret 0 is a finite subset of T and E t = E/(Θ t E ). Now suppose {U} (V xj : j<k)=. Then for eery t U, there is an integer t <kwith t/ V xt. Since E is a direct product, there is an element e E such that for eery t T 0 U, e x t (mod Θ t ). Recall that A is assumed to satisfy Sep L (c, d) witha 0,a 1,...,a r 1,b as witnesses. Since e A, this ensures that for some i<r,σ A (a i,e,α) σ A (a i,e,β). Since eery a i, e, α and β is a member of E this can be computed in E as well, thus σ E (a i,e,α) σ E (a i,e,β). Howeer terms of E are computed coordinatewise, so for any i<r,ift T 0 Uthen e x t a i (mod Θ t )(sincet / V xt ), and for t T 0 U, α β (mod Θ t ). Therefore the elements σ E (a i,e,α)andσ e (a i,e,β) agree in eery coordinate of E, somustbe equal. This is a contradiction and concludes the proof. Let us now turn to the amalgamation class. Amal(V) has proed to be a difficult class to characterize, een for ery well-behaed arieties. The aim of this paper is to show that, at least for discriminator arieties, the class has a ery satisfactory description, namely by a set of first order sentences. 1] is an in-depth study of the subject and contains the characterization of Amal(V) that will sere as the starting point here. 4

5 u w w u w Definition 5. Let V be a ariety, A V. We define (1) V asi =Amal(V) V si ; (2) A $ = (A/Ψ :Ψ Con A and A/Ψ V asi ); (3) µ A is the canonical homomorphism from A to A $. Werner proed 7, Theorem 2.2(11)] that eery discriminator ariety is filtral, hence has the congruence extension property. A maximal simple algebra of V is a simple algebra of V with no proper, simple extensions in V. Theorem 6. 1, 3.5 and 4.9] Let V be a finitely generated discriminator ariety. (1) For A V si, A V asi if and only if for eery pair of maximal simple algebras B 0, B 1 extending A, there is an isomorphism of B 0 with B 1 which is the identity on A. (2) For A V,A Amal(V) if and only if for eery maximal simple algebra M and homomorphism λ: A M, there is a homomorphism λ: A $ M such that λ µ A = λ. Corollary 7. Let V be a finitely generated discriminator ariety, A V. Then A Amal(V) if and only if (i) µ A is one-to-one, and (ii) for eery maximal simple M, eeryθ Con(A) and η embedding A/Θ into M, there exists Θ Con(A $ ) and η embedding A $ / Θ into M such that Θ A =Θand η (µ/ Θ) = η. (Hereµ/ Θ: A/Θ A $ / Θ takes a/θ to µ(a)/ Θ.) Proof. The following diagram should suggest the proof with Θ = ker λ. A $ N N P N µ A A A $ / Θ ] µ/ Θ A/Θ Suppose V is of finite type, K, L are simple algebras of V and ν is an embedding of K into L. Write K = {k 0,k 1,...,k r 1 }. By an argument almost identical to the one preceding Theorem 4, there are terms γ and δ so that the formula Fac K (x 0,x 1,...,x r 1,u,)gienby ( y)( z) γ( x, y, u, ) δ( x, y, u, ) ( σxi,z,γ( x, y, u, )] σx i,z,δ( x, y, u, )] )] i<r is such that A Fac K ( a, c, d) if and only if there is a coatom Ψ of Con(A) such that c d (mod Ψ) and {a 0 /Ψ,...,a r 1 /Ψ} =K with a j /Ψ being mapped to k j for all j<r. η η ] M 5

6 Similarly, there is a formula Ext L,ν,K ( x, y, u, ) such that A Ext L,ν,K ( a, b, c, d) if and only if there is a coatom Ψ of Con(A) such that c d (mod Ψ), A/Ψ ={a 0 /Ψ,...,a r 1 /Ψ,b 0 /Ψ,...,b s 1 /Ψ} and there is an isomorphism of A/Ψ withlwhich carries a j /Ψtoν(k j ) for all j<r. Theorem 8. Let V be a finitely generated discriminator ariety of finite type. Then Amal(V) is finitely axiomatizable. Proof. Begin with the set Σ axiomatizing Ps(V asi ) proided by Theorem 5. Then A Σ if and only if µ A is one-to-one. To apply Corollary 7, let M be maximal simple, Θ Con(A) andηan embedding of A/Θ intom.sincevhas the congruence extension property, K = A/Θ is simple. We need a sentence equialent to the existence (in the presence of Σ )of ηand Θ as in 7(ii). Since V is finitely generated, there are only finitely many pairs (up to isomorphism) L i,ν i such that: (i) L i V asi, (ii) ν i is an embedding of K into L i and (iii) there exists τ : L i M such that τ ν i = η. Leti=0,1,...,m 1enumerate those pairs and let P K,η be the sentence ( x) ( u, ) Fac K ( x, u, ) ( y) Ext Li,ν i,k( x, y, u, ) ]. m 1 i=0 We erify that if A Σ {P K,η } then Θ and ηexist with the desired properties. The conerse is left to the reader. Choose a sequence a 0,a 1,...,a r 1 of coset representaties for A by Θ. Since A/Θ =K, for any (c, d) ΘwehaeA Fac K ( a, c, d). Therefore, by assumption there is an i<msuch that A ( y) Ext Li,ν i,k( a, y,c,d), wheneer (c, d) Θ. Let T = { Ψ Con(A) :A/Ψ V asi }. Let U consist of those Ψ T such that there is an isomorphism of A/Ψ withl i taking a j /Ψtoν i (k j ), for all j<r.also, for each pair (c, d) inψ,letv(c, d) ={Ψ T:(c, d) Ψ }. Consider the family F = {U} {V(c, d) :(c, d) Θ }. We claim that F is contained in an ultrafilter oer T. To show this, it suffices to check the finite intersection property. So, let p<ωand (c 0,d 0 ),...,(c p,d p ) be pairs from Θ. In a discriminator ariety, eery compact congruence is principal (see 7, 2.2.(8)]) so there exists c, d A such that for eery Ψ Con(A), (c, d) Ψ if and only if (c j,d j ) Ψ, for all j p. In particular, (c, d) ΘsoA ( y) Ext Li,ν i,k( a, y,c,d). By the ery construction of the formula Ext, there is a congruence Ψ contained in {U} V(c, d) and hence in each V (c j,d j ), j p. Since p as well as the (c j,d j )s were arbitrary, F has the finite intersection property. Thus there is an ultrafilter D oer T containing these sets. Set Θ =Das a congruence on A $. Since eery V (c, d) D, Θ A Θ. Since U D, we get the opposite inclusion as well as an isomorphism ξ of A $ / Θ withl i whose composition with µ/ Θ equals ν i. Setting η to be τ ξ (the map associated with (L i,ν i )) one erifies that η (µ/ Θ) = η. (See diagram.) Finally, the proof can be completed by obsering that there are only finitely many pairs (K,η) (up to isomorphism) such that η is an embedding of K into a maximal simple algebra. Define P to be the formula P K,η, the conjunction oer all such pairs. Then the set Σ {P}axiomatizes Amal(V). 6

7 u w w u u w A $ ] µ A A A $ / Θ ξ = Li ] µ/ Θ A/Θ = K ν i η τ ] M A. careful examination of the sentences inoled will reeal that the characterizations in Theorems 5 and 8 are in complexity. It is not hard to show that for any ariety V, Amal(V) is closed under unions of chains (take an ultraproduct). Thus, by the Chang- Los-Susko theorem, there is an axiomatization which is in complexity. This can be achieed by omitting the subformulas ( z) σ(x, z, α) σ(x, z, β) ] from Fac, Ext, and Sep. Since the proofs are more complicated, we hae not taken that tack. Is a similar reduction possible for Ps(S)? References 1. C. Bergman, The amalgamation classes of some distributie arieties, Algebra Uniersalis 20 (1985), S. Burris and H. P. Sankappanaar, A Course in Uniersal Algebra, Springer- Verlag, New York, G. Grätzer, Uniersal Algebra, 2nd ed., Springer-Verlag, New York, B. Jónsson, Algebras whose congruence lattices are distributie, Math. Scand. 21, R. McKenzie, On spectra, and the negatie solution of the decision problem for identities haing a finite nontriial model, J. Symbolic Logic 40 (1975), A. F. Pixley, Functionally complete algebras generating distributie and permutable classes, Math. Z. 114 (1970), H. Werner, Discriminator Algebras, Studium zur Algebra und ihre Andwendungen 6, Akademie Verlag, Berlin, Uniersity of Hawaii at Manoa, Honolulu, HI