# Discrimination in a General Algebraic Setting

Authors: Benjamin Fine; Anthony Gaglione; Seymour Lipschutz; Dennis Spellman
Journal: The Scientific World Journal
 (2015)

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## Abstract

Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.


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## Body

## 1. Introduction

The notion of adiscriminating group (distinct from an older notion due to Neumann [1]) was introduced by Baumslag et al. in [2] as an outgrowth of the theory of algebraic geometry over groups. A more general class of groupstermed squarelike groups was introduced in [3] by Fine et al. This class was subsequently shown to be the axiomatic closure of the class of discriminating groups [4]. A complete overview of discriminating groups can be found in [3]. In [5] Belegradek observed that these notions should be universal, in the sense of universal algebra, and hence the analogues of the definitions and the proofs of many of the theorems go through in a general algebraic context. In this paper we explicitly carry this out.The paper has five sections. In an effort to make this paper relatively self contained, we develop in Section2 universal algebra and in Section 3 we present an overview of the logic and model theory we must apply. Section 4 is the heart of the paper and gives the development of discrimination in a universal algebraic context. Finally in Section 5 we use some results in Section 4 to provide a different proof of a classical theorem of Malcev on axiomatic classes of Ω-algebras (see [6]).


## 2. Universal Algebra

An operator domainΩ is an ordered triple (FΩ,CΩ,dΩ) where FΩ and CΩ are disjoint sets (possibly empty) and dΩ is a function from FΩ into the set of positive integers. FΩ is the set of function symbols of Ω, CΩ is the set of constant symbols of Ω, and dΩ is the degree function or arity function of Ω. Given an operator domain Ω, an Ω-algebra A is an ordered triple (1)A,fAf∈FΩ,cAc∈cΩ,where(1)
A is a nonempty set (the domain or carrier or universe of A);(2)
for eachc∈CΩ, cA∈A;(3)
for eachf∈FΩ with dΩ(f)=n, fA:An→A is an n-ary operation defined on A.AnΩ-algebra whose carrier is a singleton is called trivial. All other Ω-algebras are nontrivial. If A and B are Ω-algebras, then A is a subalgebra of B provided that(1)
A⊆B;(2)
for eachc∈CΩ, cA=cB;(3)
for eachf∈FΩ with dΩ(f)=n, fA=fB|An.LetA and B be Ω-algebras. A function ϕ:A→B is a homomorphism provided that(1)
for eachc∈CΩ, ϕ(cA)=cB;(2)
for eachf∈FΩ with dΩ(f)=n and all (a1,…,an)∈An, ϕ(fA(a1,…,an))=fB(ϕ(a1),…,ϕ(an)).Epimorphism, monomorphism, isomorphism, endomorphism, and automorphism are defined in the obvious way. LetA be an Ω-algebra. An equivalence relation R on A is a congruence on A provided for all f∈FΩ with dΩ(f)=n and all (2)x1,…,xn,y1,…,yn∈An×Anwith xiRyi for i=1,…,n, one has (3)fAx1,…,xnRfAy1,…,yn.IfR is a congruence on A, then the quotient set A/R={[a]R;a∈A} can be made into a Ω-algebra B=A/R by setting cB=[cA]R for all c∈CΩ and defining (4)fBa1R,…,anR=fAa1,…,anRfor all f∈FΩ with dΩ(f)=n and all (a1,…,an)∈An. The function A→A/R given by a↦[a]R is an epimorphism A→A/R.Ifϕ:A→B is a homomorphism, then the image of ϕ is a subalgebra of B; moreover, the relation Ker(ϕ) is defined by xKer(ϕ)y if and only if ϕ(x)=ϕ(y) is a congruence on A and A/Ker(ϕ) is isomorphic to the image of ϕ. Let (Ai)i∈I be an indexed family of Ω-algebras. Let (5)Ai=Ai,fif∈FΩ,cic∈CΩfor all i∈I. Let P=∏i∈I‍Ai. We make P into an Ω-algebra (6)P=∏i∈I‍Aiby setting cP(i)=ci for all i∈I and defining(7)fPα1,…,αni=fiα1i,…,αnifor all f∈FΩ with dΩ(f)=n, all (α1,…,αn)∈Pn, and all i∈I. P is the direct product of the family (Ai)i∈I. For each fixed i0∈I the projection πio:P→Ai0 given by α↦α(i0) is an epimorphism. If all the Ai are the same algebra A, then P=AI is a direct power of A. In that event the diagonal map δ:A→AI given by a↦δ(a) defined by δ(a)(i)=a for all a∈A and all i∈I is a monomorphism.A nonempty class ofΩ-algebras closed under taking subalgebras, homomorphic images, and direct products is a variety of Ω-algebras. Note that since the trivial Ω-algebra is a homomorphic image of any Ω-algebra whatsoever, every variety contains the trivial algebra.Example 1.
(1) IfFΩ={·,-1} with dΩ(·)=2 and dΩ(-1)=1 and CΩ={1}, then the class of all groups is a variety of Ω-algebras.
(2) LetR be an integral domain. If FΩ=R∪{+,-,[xx]} with dΩ(-)=dΩ(α)=1 for all α∈R and dΩ(+)=dΩ([x])=2 and CΩ={0}, then the class of all Lie algebras over R is a variety of Ω-algebras.LetP(I) be the power set of a set I. A subset M0⊆P(I) will be called an ideal in the ring P(I) if ∅∈M0 and M0 is closed under finite unions and also closed under the formation of subsets. M0 will be a proper ideal in P(I) if is an ideal in P(I) and I∉M0. The dual of a proper ideal in a Boolean algebra is a filter. Specifically, if I is a nonempty set and D is a family of subsets of I, then D is a filter on I provided that(1)
∅∉D;(2)
I∈D;(3)
(A,B)∈D2 implies A∩B∈D;(4)
A⊆B⊆I and A∈D implies B∈D.IfP=∏‍Ai is the direct product of the indexed family (Ai)i∈I of Ω-algebras and D is a filter on I, then we may define a congruence R(D) on P by α≅RDβ provided {i∈I:α(i)=β(i)}∈D. The quotient algebra P/RD is the reduced product of the family (Ai)i∈I modulo the filter D on I. If P=AI is a direct power, then P/RD, written AI/D, may be called the reduced power of A modulo the filter D on I. In that event one can prove that the mapping d:A→AI/D defined by a↦[δ(a)]R(D) where δ is the diagonal embedding δ:A↦AI, δ(a)(i)=a for all a∈A, i∈I, is an algebra monomorphism.The mapd:A→AI/D is called the canonical embedding of A into the reduced power AI/D. We give as examples two extreme cases.Example 2.
(1) LetD={I} be the trivial filter on I. Then P/R(D) is isomorphic to P so that direct products (powers) may be viewed as special cases of reduced products (powers).
(2) A maximal filterD on I is called an ultrafilter on I. If D is an ultrafilter on I, then P/R(D)(AI/D) is called the ultraproduct of the family (Ai)i∈I (of the algebra A) modulo the ultrafilter D on I. If P=AI is a direct power then the ultraproduct is called an ultrapower.Now view the setω of nonnegative integers with its natural order as the first limit ordinal. Suppose A0 is an Ω-algebra and, for each n<ω, Dn is an ultrafilter on an index set In. Let An+1=AnIn/Dn if An has already been defined and let dn:An→An+1 be the canonical embedding. Then the direct limit Aω of the system (8)A0→doA1→d1⋯An⋯is called the ultralimit of Ao with respect to the sequence (Dn)n<ω of ultrafilters. Note that in this event the limit map dω:A0→Aω is a monomorphism.


## 3. Model Theory and Logic

To each operator domainΩ there corresponds the first order language with equality LΩ. Besides Ω we need a countably infinite set {xn:n<ω} of distinct variables. The terms or polynomials or words of LΩ are defined recursively as follows.(1)
Constant symbols and variables are terms.(2)
Iff∈FΩ and dΩ(f)=n and (t1,…,tn) is a tuple of terms already defined, then f(t1,…,tn) is a term.An expression of the formt1=t2 where (t1,t2) is an ordered pair of terms of LΩ is an atomic formula of LΩ. The negation ~(t1=t2) of an atomic formula of LΩ is a negated atomic formula of LΩ. If α is either an atomic formula or a negated atomic formula of LΩ, then α is a literal of LΩ. We omit the recursive definition of (general) formula of LΩ but appeal to the classical result that every formula of LΩ is logically equivalent to one in prenex normal form. Such a formula is of the form Q1y1…Qnynϕ where each yi=xni is a variable, each Qi is a quantifier, ∀,∃, and ϕ, thematrix of the formula, is a Boolean combination of atomic formulas. We do not exclude the possibility n=0. Such formulas are quantifier free. Each quantifier free formula of LΩ is equivalent to a quantifier free formula of LΩ in disjunctive normal form (i.e., a disjunction of conjunctions of literals) as well as to a quantifier free formula of LΩ in conjunctive normal form (i.e., a conjunction of disjunctions of literals). A formula of LΩ containing no unquantified variables is a sentence of LΩ. We omit the recursive definition of a formula Ψ of LΩ holding in an Ω-algebra A under an interpretation of the variables and trust the reader to understand intuitively what it means for a sentence a of LΩ to hold in an Ω-algebra A. If in Q1y1…Qnyn all the Qi are ∀, then the above formula is a universal formula of LΩ. Similarly, if all the Qi are ∃, then the above formula is an existential formula of LΩ. A universal (existential) formula of LΩ containing no unquantified variables is a universal (existential) sentence of LΩ. Clearly, the negation of a universal sentence is logically equivalent to an existential sentence and vice versa. Vacuous quantifications are permitted. Thus, a quantifier free formula of LΩ is considered a universal formula of LΩ as well as an existential formula of LΩ. For example, the formula 1·1-1=1 in the language of group theory is considered both a universal sentence and an existential sentence.An existential sentence of the form(9)∃y¯∧iϕiy¯,where y¯ is a tuple of distinct variables and each ϕi(y¯) is a literal of LΩ containing at most the variables in y¯, is a primitive sentence of LΩ. Clearly a universal sentence of LΩ of the form (10)∀y¯∧iϕiy¯,where y¯ is a tuple of distinct variables and each ϕi(y¯) is a literal of LΩ containing at most the variables in y, is equivalent to the negation of a primitive sentence of LΩ. We will find it convenient to call such sentences negated primitive. Given an Ω-algebra A, we let Th∀(A) be the set of all universal sentences of LΩ true in A, Th∃(A) the set of all existential sentences of LΩ true in A, and Th(A) the set of all sentences of LΩ true in A. We call Th∀(A) the universal theory of A, Th∃(A) is the existential theory of A and Th(A) the theory of A. Clearly Th∀(A)=Th∀(B) if and only if Th∃(A)=Th∃(B). In that event we say that A and B are universally equivalent and write A≡∀B. If Th(A)=Th(B) we say that A and B are elementarily equivalent and write A≡B. Being elementarily equivalent is a sufficient but not necessarily necessary condition for being universally equivalent.Suppose∃y¯(∨iΨi(y¯)) is an existential sentence of LΩ whose matrix ∨iΨi(y¯) is written in disjunctive normal form (so that each Ψi(y¯) is a conjunction of literals). Then the above is logically equivalent to the disjunction ∨i∃Ψi(y¯)) of primitive sentences of LΩ. Similarly, if ∀y¯(∧iΨi(y¯)) is a universal sentence of LΩ whose matrix ∧iΨi(y¯) is written in conjunctive normal form (so that each Ψi(y¯) is a disjunction of literals), then it is equivalent to the conjunction ∧i∀y¯Ψi(y¯) of negated primitive sentences of LΩ.Suppose that theΩ-algebra A is a subalgebra of the Ω-algebra B. Then every existential sentence of LΩ holding in A holds also in B and every universal sentence of LΩ holding in B holds also in A. The next result is rather obvious. We omit a proof.Proposition 3.
Let theΩ-algebra A be a subalgebra of the Ω-algebra B. Then the following three statements are equivalent in pairs:(1)
A≡∀B;(2)
every primitive sentence ofLΩ true in B is also true in A;(3)
every negated primitive sentence ofLΩ true in A is also true in B.Now letQ1y1…Qnyn(∧iΨi(y¯)) be a sentence of LΩ whose matrix ∧iΨi(y¯) is written in conjunctive normal form (so each Ψi(y¯) is a disjunction of literals). If in each conjunct Ψi(y¯) at most one disjunct is atomic, then the above sentence is a Horn sentence of LΩ.A universal Horn sentence ofLΩ has the form ∀y¯Ψi(y¯) where in each conjunct Ψi(y¯) at most one disjunct is atomic. This is equivalent to the conjunction of negated primitive Horn sentences ∧i∀y¯Ψi(y¯) so we focus on negated primitive Horn sentences ∀y¯Ψ(y¯) where Ψ(y¯) is a disjunction of literals and at most one disjunct is atomic. Assume first that exactly one disjunct is atomic. Then (abbreviating ~(si=ti) as (si≠ti)) Ψ(y¯) has the form ∨i(si≠ti)∧(s=t) so that ∀y¯Ψ(y¯) is equivalent to the quasi-identity ∀y¯(∧i(si=ti)→(s=t)). The special case when Ψ(y¯) contains exactly one atomic formula but no negated atomic formulas is the identity or law ∀y¯(s=t).It follows from this discussion that ifA is an Ω-algebra, then we have the inclusions (11)IA⊆QA⊆HA⊆Th∀A,where I(A) is the set of identities of LΩ true in A, Q(A) is the set of quasi-identities of LΩ true in A, and H(A) is the set of universal Horn sentences of LΩ true in A. (Here and subsequently we commit the innocuous abuse of identifying a quasi-identity with the universal Horn sentence to which it is logically equivalent.) An Ω-algebra A is a model of a set S of sentences of LΩ provided every sentence s∈S holds in A. Appealing to the classical Godel-Henkin Completeness Theorem, we see that a set S of sentences of LΩ is consistent if and only if it has a model. Let S be a consistent set of sentences of LΩ. Then M(S) will be the class of all models or the model class of S.A class ofΩ-algebras is axiomatic provided that it is the model class M(S) for at least one consistent set S of sentences of LΩ. Every axiomatic class of Ω-algebras is nonempty and closed under isomorphism. Note that every set of quasi-identities (so, in particular, every set of identities) of LΩ holds in the trivial Ω-algebra, hence it is consistent. A celebrated theorem of Garrett Birkhoff asserts that a class of Ω-algebras is a variety if and only if it is the model class of a set of identities of LΩ. If we define a quasivariety of Ω-algebras to be an axiomatic class of Ω-algebras containing the trivial Ω-algebra and closed under taking subalgebras and direct products, then a well-known characterization, due to Mal’cev, along the lines of Birkhoff’s Theorem asserts that a class of Ω-algebras is a quasivariety if and only if it is the model class of a set of quasi-identities of LΩ. Note that the model class operator M applied to sets of sentences reverses inclusions.Now letA be an Ω-algebra. Recall that (12)IA⊆QA⊆HA⊆Th∀Ahere I(A) is the set of identities of LΩ true in A, Q(A) is the set of quasi-identities of LΩ true in A and H(A) is the set of universal Horn sentences of LΩ true in A, and Th∀(A) is the set of universal sentences of LΩ true in A. (All of these sets are consistent since they have A as a model.) Applying the model class operator, we get (13)MIA⊇MQA⊇MHA⊇MTh∀A.The set M(I(A))=var(A) is the least variety of Ω-algebras containing A. The set M(Q(A))=qvar(A) is the least quasivariety of Ω-algebras containing A. The set M(H(A))=uhc(A) is the least universally axiomatizable Horn class containing A. Finally, M(Th∀(A)) (the universal closure of A denoted by ucl(A)) is the least universally axiomatizable class containing A. With the above notation, we have (14)uclA⊆uhcA⊆qvarA⊆varA.A monomorphismϵ:A→B is an elementary embedding provided for each formula Ψ(y1,…,yn) and each tuple (a1,…,an) from A it is the case that Ψ(ϵ(a1),…,ϵ(an)) holds in B if and only if Ψ(a1,…,an) holds in A. The existence of an elementary embedding is a sufficient, but in general unnecessary, condition for elementary equivalence. If A is a subalgebra of B and the inclusion map embeds A elementarily into B, then B is said to be an elementary extension of A. Obviously isomorphic Ω-algebras are elementarily equivalent. Thus, necessary conditions for a nonempty class X of Ω-algebras to be axiomatic are that X be closed under elementary equivalence and taking ultraproducts. These conditions are also sufficient. That is the content of Theorem 3, Section 42 of [7] (see [8]).LetS be a consistent set of sentences of LΩ. A consequence of a theorem of Los is that if every member of the family (Ai)i∈I of Ω-algebras lies in M(S), then so does every ultraproduct of the family. From that it is easy to deduce that every ultrapower and every ultralimit of an Ω-algebra A must be elementarily equivalent to A. (If some sentence α true in U an ultrapower or ultralimit of A were false in A, then its negation ~α would be true in A. But then both α and ~α would hold in U, an explicit contradiction.) The canonical embedding is in fact an elementary embedding (see, e.g., [8]).Proposition 4.
Letσ be a sentence of LΩ.(1)
σ is equivalent to a universal sentence of LΩ if and only if σ is preserved under taking subalgebras.(2)
σ is equivalent to a Horn sentence of LΩ if and only if σ is preserved under taking reduced products.(3)
Ifσ is a universal sentence of LΩ, then σ is equivalent to a universal Horn sentence of LΩ if and only if σ is preserved under taking direct products.For proofs see, for example, [7].We will have need in the next section to consider special varieties ofΩ-algebras under a restriction on the operator domain Ω. Specifically, we consider those Ω whose set CΩ of constant symbols is a singleton CΩ={θ}. We then consider those varieties of Ω-algebras satisfying, at least, the laws f(θ,…,θ)=θ as f varies over the set FΩ of function symbols of Ω. Recall that quantifier free sentences of LΩ are special cases of universal sentences of LΩ. We say that such varietiescontain a zero. For example, the varieties of groups, monoids, and Lie algebras contain a zero but the variety of semigroups does not. We conclude this section with several observations.Observation 1.
Suppose theΩ-algebra A is a subalgebra of the Ω-algebra B. Since A≡∀B if and only every primitive sentence of LΩ true in B is also true in A it follows that necessary and sufficient conditions for A≡∀B are that every finite system (in finitely many variables) of equations and inequations (15)Pi=pi,i∈I,Qj≠qj,j∈J,where the Pi,pi,Qj, and qj are terms of LΩ, which has a solution in B must already have a solution in A.Observation 2.
A negated primitive Horn sentence∀y¯(∨i(si≠ti)) whose matrix ∨i(si≠ti) contains no atomic formula is false in the trivial Ω-algebra. Hence, if such a sentence is true in an Ω-algebra A, then uhc(A), the universal Horn class of A, must be a proper subclass of qvar(A), the quasivariety generated by A. On the other hand, if A lies in a variety containing a zero θ, then A contains the trivial subalgebra {θA} so such a sentence could not hold in A.Observation 3.
Suppose theΩ-algebra B is a direct power of the Ω-algebra A. For some special A we will want to show that A and B are universally equivalent. Since the diagonal map δ:A→B embeds A isomorphically into B every universal sentence of LΩ true in B must also be true in A. Thus, in the above situation it will suffice for our purposes to show that every universal sentence of LΩ true in A is also true in B.


## 4. Discriminating and Squarelike Algebras

LetΩ be an operator domain and let V be a variety of Ω-algebras. Throughout this section we will assume that all algebras lie in V. We will sometimes (but not always) assume that V contains a zero, which in that event implies, among other things, a restriction on Ω.Definition 5.
LetA and B be elements of V.(1)
A separates B provided to every pair x≠y of unequal elements of B there is a homomorphism ϕ:B→A such that ϕ(x)≠ϕ(y).(2)
A discriminates B provided given finitely many pairs xi≠yi, i=1,…,n, of unequal elements of B there is a homomorphism ϕ:B→A such that ϕ(xi)≠ϕ(yi) for all i=1,…,n.(3)
A is discriminating provided it discriminates every element of V which it separates.Theorem 6.
A is discriminating if and only if it discriminates its direct square A2.The proof is identical to that for groups. See, for example, [9].Theorem 7.
IfA is discriminating, then A2≡∀A; that is, A has the same universal theory as its direct square A2.Proof.
We identifyA with its image in A2 under the diagonal embedding δ:A→A2 given by δ(a)=(a,a). Viewing A as a subalgebra of A2, it will suffice to show that every finite system (16)Pix1,…,xn=pix1,…,xn,i∈IQjx1,…,xn≠qjx1,…,xn,j∈Jof equations and inequations having a solution in A2 already has a solution in A.
Suppose that(x1,…,xn)=(b1,…,bn) is a solution in A2 to the above system. Since A discriminates A2 there is a homomorphism ϕ:A2→A such that (17)ϕQjb1,…,bn=Qjϕb1,…,ϕbn≠ϕqjb1,…,bn=qjϕb1,…,ϕbn,j∈J.
But then(x1,…,xn)=(ϕ(b1),…,ϕ(bn)) is a solution to the system in A. Hence, A2≡∀A.Definition 8.
An algebraA in V is squarelike provided A2≡∀A; that is, A has the same universal theory as its direct square A2.Thus, every discriminating algebra is squarelike.Theorem 9.
LetA be an algebra in V. The following three conditions are equivalent in pairs:(1)
A is squarelike;(2)
ucl(A)=uhc(A);(3)
there is a discriminating algebraB in V such that A≡∀B.Momentarily assuming the theorem, we have the following consequence.Corollary 10.
Suppose the varietyV contains a zero and let A be an algebra in V. Then the following three conditions are equivalent in pairs:(1)
A is squarelike;(2)
ucl(A)=qvar(A);(3)
there is a discriminating algebraB in V such that A≡∀B.Proof of Corollary.
Assuming the theorem it will suffice to show that uhc(A) = qvar(A). Since uhc(A) is axiomatizable by universal Horn sentences it is closed under taking subalgebras and direct products. Since A∈uhc(A) and the trivial algebra {θA} is a subalgebra of A, uhc(A) contains the trivial algebra. Thus the axiomatic class uhc(A) is closed under taking subalgebras and direct products and contains the trivial algebra. Hence, it is a quasivariety. Therefore, uhc(A) = qvar(A).We begin the proof of Theorem9 with a sequence of lemmas.Lemma 11.
LetA and B be Ω-algebras. Then every universal sentence of LΩ true in B is also true in A if and only if A embeds monomorphically in an elementary extension *B of B.This is Theorem 3, Chapter 7, Section 43 of [7].Lemma 12.
Direct products and reduced products preserve elementary equivalence.This is Theorem6.3.4. of [10].Lemma 13.
IfA is squarelike, then, for every integer n≥2, An≡∀A.Proof.
We use induction onn. The result holds for n=2. Now suppose the result holds for n=k. Thus, every universal sentence of LΩ true in A is also true in Ak. By Lemma 11 there is an elementary extension A⋆ of A such that Ak embeds in A⋆. Then Ak+1embeds in A⋆×A. Since A⋆≡A and A≡A we have by Lemma 12 that A⋆×A≡A2. In particular, A⋆×A≡∀A2. But A is squarelike so A2≡∀A. It follows that A⋆×A≡∀A2. Hence, every universal sentence of LΩ true in A is also true in A⋆×A. Since Ak+1 embeds in A⋆×A, every universal sentence of LΩ true in A is also true in Ak+1. Therefore, Ak+1≡∀A by Observation 3 of the previous section. That completes the induction and proves the lemma.Proof of Theorem9.
(1)⇒ (2) Assume A is squarelike. It will suffice to show that ucl(A) is axiomatizable by universal Horn sentences. Assume deducing a contradiction that u is a universal sentence of LΩ true in A but not a consequence of any set of universal Horn sentences of LΩ. Of course then u itself cannot be a Horn sentence. We may assume that the matrix of u is written in conjunctive normal form and hence u has the form ∀y¯(∧iϕi(y¯)) where each ϕi(y¯) is a disjunction of literals. Thus, u is equivalent to the conjunction ∧i∀y¯ϕi(y¯) of negated primitive sentences ∀y¯ϕi(y¯). Furthermore, at least one ϕi(y¯) must contain at least two atomic disjuncts or else u would be a Horn sentence. We claim that at least one of the negated primitive sentences ∀y¯ϕi(y¯) containing at least two atomic disjuncts cannot be a consequence of any set of universal Horn sentences of LΩ true in A. Suppose not. For each i such that ϕi(y¯) contains at least two atomic disjuncts let Hi be a set of universal Horn sentences of LΩ true in A such that ∀y¯ϕ(y¯) is a consequence of Hi. For each i for which ϕi(y¯) contains at most one atomic disjunct, ∀y¯(ϕi(y¯)) is already a universal Horn sentence. For such i we take Hi={∀y¯ϕi(y¯)} and let H=∪iHi. It follows then that u would be a consequence of the set H of universal Horn sentences of LΩ true in A, contrary to hypothesis. The contradiction shows that at least one negated primitive sentence ∀y¯ϕi(y¯) for which ϕi(y¯) contains at least two atomic disjuncts cannot be a consequence of any set of universal Horn sentences of LΩ true in A.
Fix such a conjunct∀y¯ϕ(y¯) (we suppress i notationally). Let ϕ(y¯) be (18)⋁μ=1mQμy¯≠qμy¯∨⋁ν=1nPνy¯=pνy¯,where n≥2. For each fixed ν0=1,…,n let Ψν0 be the quasi-identity (19)∀y¯⋀μ=1mQμy¯=qμy¯⟶Pν0y¯=pν0y¯.
Suppose deducing a contradiction thatΨν0 is true in A. Then for every tuple y¯=a¯ from A, (20)⋀μ=1mQμa¯=qμa¯⟶Pν0a¯=pν0a¯would be true in A. Equivalently, for every tuple y¯=a¯ from A, (21)⋁μ=1mQμa¯≠qμa¯∨Pν0a¯=pν0a¯would hold in A. Since a disjunction is true if at least one of its disjuncts is true we would have in the above event that (22)⋁μ=1mQμa¯≠qμa¯∨⋁ν=1nPνa¯=pνa¯holds in A for every tuple y¯=a¯ from A. This implies that ∀y¯ϕ(y¯) would be a consequence of the quasi-identity Ψν0. But quasi-identities are special cases of universal Horn sentences and ∀y¯ϕ(y¯) was postulated not to be a consequence of any set of universal Horn sentences of LΩ true in A. The contradiction shows that none of the quasi-identities Ψν can hold in A. Thus, for each ν=1,…,n, the existential sentence (23)∃y¯⋀μ=1mQμy¯=qμy¯∨Pνy¯≠pνy¯holds in A.
Now lety¯=a¯ be a tuple of elements from A such that simultaneously Qμ(aν¯)=qμ(aν¯) for all μ=1,…,m and Pν(aν¯)≠pν(aν¯). Let a¯=(a1¯,…,an¯) be the tuple from An so that (24)⋀μ=1mQμa¯=qμa¯∧⋀ν=1nPνa¯≠pνa¯holds in An. By Lemma 13, An≡∀A. It follows that the existential sentence (25)∃y¯⋀μ=1mQμy¯=qμy¯∧⋀ν=1nPνy¯≠pνy¯holds in A. But that contradicts the fact that its negation (up to logical equivalence) ∀y¯ϕ(y¯) holds in A. The contradiction shows that ucl(A) must have at least one set of universal Horn axioms. Hence, ucl(A)=uhc(A) if A is squarelike.
(2)⇒ (3) Suppose ucl(A)=uhc(A). Now A∈uhc(A) and uhc(A) is closed under taking direct products. Let I be an infinite index set and let B=AI. Then B∈ucl(A) so B is a model of Th∀(A) and every universal sentence of LΩ true in A is also true in B. Thus, A≡∀B by Observation 3 of the previous section. Now B2 is isomorphic to B so that B is discriminating by Theorem 6.
(3)⇒ (1) Suppose that A≡∀B where B is discriminating. Then, in particular, every universal sentence of LΩ true in B must also be true in A. Thus, by Lemma 11, A embeds in an elementary extension B⋆ of B. Now A2 embeds in (⋆B)2 which is elementarily equivalent to B2 by Lemma 12. In particular, (⋆B)2≡∀B2. But B is discriminating so B2≡∀B by Theorem 7. Now A≡∀B so ultimately (⋆B)2≡∀A. Thus, every universal sentence of LΩ true in A is also true in (⋆B)2. But A2 embeds in (⋆B)2 so every universal sentence of LΩ true in A must also be true in A2. Then A2≡∀A by Observation 3 of the previous section. That is, A is squarelike.Exactly as for groups (see, e.g., [3]) we have the following.Theorem 14.
Let(Ai)i∈I be an indexed family of Ω-algebras and let D be a filter on I. Let A be the reduced product P/R(D) where P=∏i∈I‍Ai. Then A2 is isomorphic to the reduced product of the family (Ai2)i∈I modulo the filter D on I.Corollary 15.
The class of squarelike algebras inV is axiomatic.Proof.
LetX denote the class of squarelike algebras. It will suffice to show that X is closed under elementary equivalence and ultraproducts. Suppose A lies in X and B≡A. then, by Lemma 12, B2≡A2. In particular, B≡∀A and B2≡∀A2. But A∈X so A2≡∀A. It follows that B2≡∀B so that X is closed under elementary equivalence. Now let (Ai)i∈I be a family of Ω-algebras and let D be an ultrafilter on I. Let A be the ultraproduct constructed from this data. If ϕ is a sentence of LΩ let the support of ϕ be the set Supp(ϕ) of all i∈I such that ϕ holds in Ai. A consequence of Los’ Theorem (see, e.g., [8]) is that ϕ holds in A if and only if Supp(ϕ)∈D. Now consider A2. By Theorem 14, A2 is isomorphic to the ultraproduct of the family (A2)i∈I modulo the filter D on I. Suppose that each Ai lies in X so that (A2)i≡∀Ai for all i∈I. Let ϕ be a universal sentence of LΩ holding in A. Then Supp(ϕ)∈D. Thus, for each i∈Supp(ϕ), ϕ holds in Ai. But Ai≡∀(A2)i since Ai∈X. But then ϕ holds in A2 since Supp(ϕ)∈D. It follows that every universal sentence of LΩ true in A must also be true in A2 and hence A2≡∀A by Observation 3 of the previous section. So X is closed under taking ultraproducts as well as elementary equivalence. Hence, X is axiomatic.Theorem 16.
LetA be a squarelike algebra in V. Then A is elementarily equivalent to a discriminating member of V. Consequently, the class of squarelike algebras in V is the least axiomatic class containing the discriminating members of V.A proof which uses the ultralimit construction is exactly the same as that for groups and may be found, for example, in [4]. See also [5] for a different proof.Corollary 17.
The class of squarelike algebras inV has a set of Horn axioms.A theorem of Los and Suszko asserts that a model class has a set of universal-existential axioms if and only if it is closed under direct unions. For definitions of the relevant terms and a proof of the Los-Suszko Theorem see, for example, [7]. It is easy to show that the class of squarelike algebras in V is closed under taking direct unions. Thus, that class, in addition to having a set of Horn axioms, has a set of universal-existential axioms.Proof of Corollary17.
Exactly as for groups (see, e.g., [3]) one proves that a reduced product of discriminating algebras is discriminating. Now let (Ai)i∈I be a family of squarelike algebras in V and let D be a filter on I. By the theorem there is, for each i∈I, a discriminating algebra Bi in V such that Ai≡Bi. By Lemma 12 the reduced product of the family (Ai)i∈I modulo the filter D on I is elementarily equivalent to the reduced product of the family (Bi)i∈I modulo the filter D on I. In particular, the reduced product of the family (Ai)i∈I modulo the filter D on I is universally equivalent to the reduced product of the family (Bi)i∈I modulo the filter D on I. Hence, the reduced product of the family (Ai)i∈I is squarelike by Theorem 9. It follows that the class of squarelike algebras in V is preserved under taking reduced products. Now, as mentioned in page 225 of [11], an axiomatic class closed under reduced products has a set of Horn axioms. That completes the proof.In the special case whenV contains a zero θ we can explicitly describe a set of axioms for the class of squarelike algebras in V by mimicking the situation for groups. For the remainder of this section we will restrict ourselves to those varieties V which contain a zero θ. The class of squarelike algebras in the variety V containing a zero is the model class of the laws of V together with the sentences (26)⋀j∃y¯⋀iPiy¯=piy¯∧Qjy¯≠qjy¯⟶∃y¯⋀iPiy¯=piy¯∧⋀jQjy¯≠qjy¯as the Pi,pi,Qj, and qj vary over terms of LΩ containing at most the variables in y¯. See, for example, [5].Definition 18.
LetV be a variety containing a zero. Let Q be a subquasivariety of V. An algebra A∈Qq-discriminates Q provided that given finitely many quasi-identities (27)∀y¯⋀iQiy¯=qiy¯⟶Pjy¯=pjy¯with the same antecedents and none of which hold in Q there exists a tuple a¯ from A such that simultaneously Qi(a¯)=qi(a¯) and Pj(a¯)≠pj(a¯) for all i,j. An algebra A in V is q-discriminating provided it q-discriminates qvar(A).Definition 19.
LetV be a variety containing a zero. An algebra A in V is q-algebraically closed if and only if whenever a finite system (28)Piy¯=piy¯i∈I,Qjy¯≠qjy¯j∈Jof equations and inequations has a solution in some algebra B∈ qvar(A) it also has a solution in A.Exactly as for groups we have the following.Theorem 20.
LetV be a variety containing a zero. Let A be an algebra in V. The following conditions are equivalent in pairs.(1)
A is q-discriminating.(2)
A is q-algebraically closed.(3)
A is squarelike.See, for example, [12].


## 5. On a Theorem of Malcev

A classical theorem of Malcev asserts that an axiomatic class ofΩ-algebras which contains the trivial algebra and is closed under taking subalgebras and direct products must be the model class of at least one set of quasi-identities of LΩ. Granting us the result that an axiomatic class of Ω-algebra is closed under taking subalgebras if and only if it is the model class of at least one set of universal sentences of LΩ (Theorem 5.2.4 of [10]) we observe that the argument used in proving that A2≡∀A implies ucl(A)=uhc(A) can be adapted to provide a proof of Malcev’s Theorem.Theorem 21 (Malcev).
An axiomatic class ofΩ-algebras which contains the trivial algebra and is closed under taking subalgebras and direct products must be the model class of at least one set of quasi-identities of LΩ.Proof.
We assume that an axiomatic class ofΩ-algebra is closed under taking subalgebras if and only if it is the model class of at least one set of universal sentences of LΩ (Theorem 5.2.4 of [10]).
Now letτ be an axiomatic class of Ω-algebras containing the trivial algebra and closed under taking subalgebras and direct products. Then τ is the model class of a set of universal sentences of LΩ. Assume deducing a contradiction that u is a universal sentence of LΩ holding in every algebra A∈τ but that u is not a consequence of any set of quasi-identities of LΩ holding in every algebra A∈τ. We may assume that the matrix of u is written in conjunctive normal form. Hence u has the form (29)∀y¯⋀iϕiy¯,where each ϕi(y¯) is a disjunction of literals. Thus u is equivalent to the conjunction (30)⋀i∀y¯ϕiy¯of negated primitive sentences(31)∀y¯ϕiy¯.Furthermore, it must be the case that at least one ϕi(y¯) must not contain exactly one atomic disjunct or else u would be equivalent to a conjunction of quasi-identities true in every algebra A∈τ contrary to hypothesis.
We claim that it is impossible to have a conjunct(32)∀x¯⋁jQjx¯≠qjx¯whose matrix contains no atomic disjuncts. This is so since (33)∀x¯⋁jQjx¯≠qjx¯is false in the trivial algebra {Θ}∈τ. Therefore at least one ϕi(y¯) must contain at least two atomic disjuncts.
We now claim that at least one of the negated primitive sentences∀y¯ϕi(y¯) containing at least two atomic disjuncts cannot be a consequence of any set of quasi-identities of LΩ true in every algebra A∈τ. Suppose not. For each i such that ϕi(y¯) contains at least two atomic disjuncts let Hi be the set of quasi-identities of Lω true in every algebra A∈τ such that ∀y¯ϕi(y¯) is a consequence of Hi. For each i for which ϕi(y¯) contains exactly one atomic disjunct, ∀y¯(ϕi(y¯) is already (up to logical equivalence) a quasi-identity. For such i we take Hi={∀y¯ϕi(y¯)}. Let H=∪iHi. Then u would be a consequence of the set H of quasi-identities (up to logical equivalence) of LΩ true in every algebra A∈τ, contrary to hypothesis. The contradiction shows that at least one negated primitive sentence ∀y¯ϕi(y¯) for which ϕi(y¯) contains at least two atomic disjuncts cannot be a consequence of any set of quasi-identities of LΩ true in every algebra A∈τ. Fix such a conjunct ∀y¯ϕ(y¯) (we suppress i notationally). Let ϕ(y¯) be(34)⋁mu=1mQμy¯≠qμy¯∨⋁ν=1nPνy¯=pνy¯,where n≥2. For each fixed ν0=1,…,n let ψν0 be the quasi-identity (35)∀y¯⋀μ=1mQμy¯=qμy¯⟶Pν0y¯=pν0y¯.
Suppose deducing a contradiction thatψν0 is true in every algebra A∈τ. Fix an A∈τ. Then for every tuple y¯=a¯ from A(36)⋀μ=1mQμa¯=qμa¯⟶Pν0a¯=pμ)a¯would hold in A.
Equivalently, for every tupley¯=a¯ from A, (37)⋁ν=1mQμa¯≠qμa¯∨Pν0a¯=pν0a¯would hold in A. Since a disjunction is true if at least one of its disjuncts is true we have in that event that (38)⋁μ=1mQμa¯≠qμa¯∨⋁ν=1nPνa¯=pνa¯holds in A for every tuple y¯=a¯ from A. Since A was arbitrary, ∀y¯ϕ(y¯) would be a consequence of the quasi-identity ψν0 true in every algebra in τ. But ∀y¯(ϕ(y¯) was postulated not to be a consequence of any such set of quasi-identities. Thus, for each ν=1,…,n, there is an algebra Aν∈τ such that the existential sentence (39)∃y¯⋀μ=1mQμy¯=qμy¯∧Pνy¯≠pνy¯holds in Aν. Now let y¯=aν¯ be a tuple of elements from Aν such that simultaneously Qμ(aν¯)=qμ(aν¯) for all μ=1,…,m and Pν(aν¯)=pν(aν¯).
Leta¯=(a1¯,…,an¯) be the tuple from ∏ν=1n‍Aν. It follows that the existential sentence (40)∃⋀μ=1mQμy¯=qμy¯∧⋀ν=1nPνy¯≠pνy¯holds in ∏ν=1nAν∈τ.
But that contradicts the fact that its negation (up to logical equivalence)∀y¯ϕ(y¯) holds in every algebra A∈τ. The contradiction shows that τ must be axiomatizable by at least one set of quasi-identities of LΩ completing the proof.



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