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.

1. Introduction

The notion of a discriminating 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 groups termed 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 Section 2 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 -algebras (see [6]).

2. Universal Algebra

An operator domain is an ordered triple where and are disjoint sets (possibly empty) and is a function from into the set of positive integers. is the set of function symbols of , is the set of constant symbols of , and is the degree function or arity function of . Given an operator domain , an -algebra is an ordered triple where(1) is a nonempty set (the domain or carrier or universe of );(2)for each , ;(3)for each with , is an -ary operation defined on .

An -algebra whose carrier is a singleton is called trivial. All other -algebras are nontrivial. If and are -algebras, then is a subalgebra of provided that(1);(2)for each , ;(3)for each with , .

Let and be -algebras. A function is a homomorphism provided that(1)for each , ;(2)for each with and all , .

Epimorphism, monomorphism, isomorphism, endomorphism, and automorphism are defined in the obvious way. Let be an -algebra. An equivalence relation on is a congruence on provided for all with and all with for , one has

If is a congruence on , then the quotient set can be made into a -algebra by setting for all and defining for all with and all . The function given by is an epimorphism .

If is a homomorphism, then the image of is a subalgebra of ; moreover, the relation is defined by if and only if is a congruence on and is isomorphic to the image of . Let be an indexed family of -algebras. Let for all . Let . We make into an -algebra by setting for all and definingfor all with , all , and all . is the direct product of the family . For each fixed the projection given by is an epimorphism. If all the are the same algebra , then is a direct power of . In that event the diagonal map given by defined by for all and all is a monomorphism.

A nonempty class of -algebras closed under taking subalgebras, homomorphic images, and direct products is a variety of -algebras. Note that since the trivial -algebra is a homomorphic image of any -algebra whatsoever, every variety contains the trivial algebra.

Example 1. (1) If with and and , then the class of all groups is a variety of -algebras.
(2) Let be an integral domain. If with for all and and , then the class of all Lie algebras over is a variety of -algebras.

Let be the power set of a set . A subset will be called an ideal in the ring if and is closed under finite unions and also closed under the formation of subsets. will be a proper ideal in if is an ideal in and . The dual of a proper ideal in a Boolean algebra is a filter. Specifically, if is a nonempty set and is a family of subsets of , then is a filter on provided that(1);(2);(3) implies ;(4) and implies .

If is the direct product of the indexed family of -algebras and is a filter on , then we may define a congruence on by provided . The quotient algebra is the reduced product of the family modulo the filter on . If is a direct power, then , written , may be called the reduced power of modulo the filter on . In that event one can prove that the mapping defined by where is the diagonal embedding , for all , , is an algebra monomorphism.

The map is called the canonical embedding of into the reduced power . We give as examples two extreme cases.

Example 2. (1) Let be the trivial filter on . Then is isomorphic to so that direct products (powers) may be viewed as special cases of reduced products (powers).
(2) A maximal filter on is called an ultrafilter on . If is an ultrafilter on , then is called the ultraproduct of the family (of the algebra ) modulo the ultrafilter on . If is a direct power then the ultraproduct is called an ultrapower.

Now view the set of nonnegative integers with its natural order as the first limit ordinal. Suppose is an -algebra and, for each , is an ultrafilter on an index set . Let if has already been defined and let be the canonical embedding. Then the direct limit of the system is called the ultralimit of with respect to the sequence of ultrafilters. Note that in this event the limit map is a monomorphism.

3. Model Theory and Logic

To each operator domain there corresponds the first order language with equality . Besides we need a countably infinite set of distinct variables. The terms or polynomials or words of are defined recursively as follows.(1)Constant symbols and variables are terms.(2)If and and is a tuple of terms already defined, then is a term.

An expression of the form where is an ordered pair of terms of is an atomic formula of . The negation of an atomic formula of is a negated atomic formula of . If is either an atomic formula or a negated atomic formula of , then is a literal of . We omit the recursive definition of (general) formula of but appeal to the classical result that every formula of is logically equivalent to one in prenex normal form. Such a formula is of the form where each is a variable, each is a quantifier, , and , the matrix of the formula, is a Boolean combination of atomic formulas. We do not exclude the possibility . Such formulas are quantifier free. Each quantifier free formula of is equivalent to a quantifier free formula of in disjunctive normal form (i.e., a disjunction of conjunctions of literals) as well as to a quantifier free formula of in conjunctive normal form (i.e., a conjunction of disjunctions of literals). A formula of containing no unquantified variables is a sentence of . We omit the recursive definition of a formula of holding in an -algebra under an interpretation of the variables and trust the reader to understand intuitively what it means for a sentence a of to hold in an -algebra . If in all the are , then the above formula is a universal formula of . Similarly, if all the are , then the above formula is an existential formula of . A universal (existential) formula of containing no unquantified variables is a universal (existential) sentence of . 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 is considered a universal formula of as well as an existential formula of . For example, the formula in the language of group theory is considered both a universal sentence and an existential sentence.

An existential sentence of the form where is a tuple of distinct variables and each is a literal of containing at most the variables in , is a primitive sentence of . Clearly a universal sentence of of the form where is a tuple of distinct variables and each is a literal of containing at most the variables in , is equivalent to the negation of a primitive sentence of . We will find it convenient to call such sentences negated primitive. Given an -algebra , we let be the set of all universal sentences of true in , the set of all existential sentences of true in , and the set of all sentences of true in . We call the universal theory of , is the existential theory of and the theory of . Clearly if and only if . In that event we say that and are universally equivalent and write . If we say that and are elementarily equivalent and write . Being elementarily equivalent is a sufficient but not necessarily necessary condition for being universally equivalent.

Suppose is an existential sentence of whose matrix is written in disjunctive normal form (so that each is a conjunction of literals). Then the above is logically equivalent to the disjunction of primitive sentences of . Similarly, if is a universal sentence of whose matrix is written in conjunctive normal form (so that each is a disjunction of literals), then it is equivalent to the conjunction of negated primitive sentences of .

Suppose that the -algebra is a subalgebra of the -algebra . Then every existential sentence of holding in holds also in and every universal sentence of holding in holds also in . The next result is rather obvious. We omit a proof.

Proposition 3. Let the -algebra be a subalgebra of the -algebra . Then the following three statements are equivalent in pairs:(1);(2)every primitive sentence of true in is also true in ;(3)every negated primitive sentence of true in is also true in .

Now let be a sentence of whose matrix is written in conjunctive normal form (so each is a disjunction of literals). If in each conjunct at most one disjunct is atomic, then the above sentence is a Horn sentence of .

A universal Horn sentence of has the form where in each conjunct at most one disjunct is atomic. This is equivalent to the conjunction of negated primitive Horn sentences so we focus on negated primitive Horn sentences where is a disjunction of literals and at most one disjunct is atomic. Assume first that exactly one disjunct is atomic. Then (abbreviating as ) has the form so that is equivalent to the quasi-identity . The special case when contains exactly one atomic formula but no negated atomic formulas is the identity or law .

It follows from this discussion that if is an -algebra, then we have the inclusions where is the set of identities of true in , is the set of quasi-identities of true in , and is the set of universal Horn sentences of true in . (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 -algebra is a model of a set of sentences of provided every sentence holds in . Appealing to the classical Godel-Henkin Completeness Theorem, we see that a set of sentences of is consistent if and only if it has a model. Let be a consistent set of sentences of . Then will be the class of all models or the model class of .

A class of -algebras is axiomatic provided that it is the model class for at least one consistent set of sentences of . Every axiomatic class of -algebras is nonempty and closed under isomorphism. Note that every set of quasi-identities (so, in particular, every set of identities) of holds in the trivial -algebra, hence it is consistent. A celebrated theorem of Garrett Birkhoff asserts that a class of -algebras is a variety if and only if it is the model class of a set of identities of . If we define a quasivariety of -algebras to be an axiomatic class of -algebras containing the trivial -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 -algebras is a quasivariety if and only if it is the model class of a set of quasi-identities of . Note that the model class operator applied to sets of sentences reverses inclusions.

Now let be an -algebra. Recall that here is the set of identities of true in , is the set of quasi-identities of true in and is the set of universal Horn sentences of true in , and is the set of universal sentences of true in . (All of these sets are consistent since they have A as a model.) Applying the model class operator, we get The set is the least variety of -algebras containing . The set is the least quasivariety of -algebras containing . The set is the least universally axiomatizable Horn class containing . Finally, (the universal closure of denoted by ) is the least universally axiomatizable class containing . With the above notation, we have

A monomorphism is an elementary embedding provided for each formula and each tuple from it is the case that holds in if and only if holds in . The existence of an elementary embedding is a sufficient, but in general unnecessary, condition for elementary equivalence. If is a subalgebra of and the inclusion map embeds elementarily into , then is said to be an elementary extension of . Obviously isomorphic -algebras are elementarily equivalent. Thus, necessary conditions for a nonempty class of -algebras to be axiomatic are that 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]).

Let be a consistent set of sentences of . A consequence of a theorem of Los is that if every member of the family of -algebras lies in , then so does every ultraproduct of the family. From that it is easy to deduce that every ultrapower and every ultralimit of an -algebra must be elementarily equivalent to . (If some sentence true in an ultrapower or ultralimit of were false in , then its negation ~ would be true in . But then both and ~ would hold in , an explicit contradiction.) The canonical embedding is in fact an elementary embedding (see, e.g., [8]).

Proposition 4. Let be a sentence of .(1) is equivalent to a universal sentence of if and only if is preserved under taking subalgebras.(2) is equivalent to a Horn sentence of if and only if is preserved under taking reduced products.(3)If is a universal sentence of , then is equivalent to a universal Horn sentence of if and only if is preserved under taking direct products.

For proofs see, for example, [7].

We will have need in the next section to consider special varieties of -algebras under a restriction on the operator domain . Specifically, we consider those whose set of constant symbols is a singleton . We then consider those varieties of -algebras satisfying, at least, the laws as varies over the set of function symbols of . Recall that quantifier free sentences of are special cases of universal sentences of . We say that such varieties contain 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 the -algebra is a subalgebra of the -algebra . Since if and only every primitive sentence of true in is also true in it follows that necessary and sufficient conditions for are that every finite system (in finitely many variables) of equations and inequations where the , and are terms of , which has a solution in must already have a solution in .

Observation 2. A negated primitive Horn sentence whose matrix contains no atomic formula is false in the trivial -algebra. Hence, if such a sentence is true in an -algebra , then , the universal Horn class of , must be a proper subclass of , the quasivariety generated by . On the other hand, if lies in a variety containing a zero , then contains the trivial subalgebra so such a sentence could not hold in .

Observation 3. Suppose the -algebra is a direct power of the -algebra . For some special we will want to show that and are universally equivalent. Since the diagonal map embeds isomorphically into every universal sentence of true in must also be true in . Thus, in the above situation it will suffice for our purposes to show that every universal sentence of true in is also true in .

4. Discriminating and Squarelike Algebras

Let be an operator domain and let be a variety of -algebras. Throughout this section we will assume that all algebras lie in . We will sometimes (but not always) assume that contains a zero, which in that event implies, among other things, a restriction on .

Definition 5. Let and be elements of .(1) separates provided to every pair of unequal elements of there is a homomorphism such that .(2) discriminates provided given finitely many pairs , , of unequal elements of there is a homomorphism such that for all .(3) is discriminating provided it discriminates every element of which it separates.

Theorem 6. is discriminating if and only if it discriminates its direct square .

The proof is identical to that for groups. See, for example, [9].

Theorem 7. If is discriminating, then ; that is, has the same universal theory as its direct square .

Proof. We identify with its image in under the diagonal embedding given by . Viewing as a subalgebra of , it will suffice to show that every finite system of equations and inequations having a solution in already has a solution in .
Suppose that is a solution in to the above system. Since discriminates there is a homomorphism such that But then is a solution to the system in . Hence, .

Definition 8. An algebra in is squarelike provided ; that is, has the same universal theory as its direct square .

Thus, every discriminating algebra is squarelike.

Theorem 9. Let be an algebra in . The following three conditions are equivalent in pairs:(1) is squarelike;(2);(3)there is a discriminating algebra in such that .

Momentarily assuming the theorem, we have the following consequence.

Corollary 10. Suppose the variety contains a zero and let be an algebra in . Then the following three conditions are equivalent in pairs:(1) is squarelike;(2);(3)there is a discriminating algebra in such that .

Proof of Corollary. Assuming the theorem it will suffice to show that uhc() = qvar(). Since uhc() is axiomatizable by universal Horn sentences it is closed under taking subalgebras and direct products. Since and the trivial algebra is a subalgebra of , uhc() contains the trivial algebra. Thus the axiomatic class uhc() is closed under taking subalgebras and direct products and contains the trivial algebra. Hence, it is a quasivariety. Therefore, uhc = qvar().

We begin the proof of Theorem 9 with a sequence of lemmas.

Lemma 11. Let and be -algebras. Then every universal sentence of true in is also true in if and only if embeds monomorphically in an elementary extension of .

This is Theorem 3, Chapter 7, Section 43 of [7].

Lemma 12. Direct products and reduced products preserve elementary equivalence.

This is Theorem . of [10].

Lemma 13. If is squarelike, then, for every integer , .

Proof. We use induction on . The result holds for . Now suppose the result holds for . Thus, every universal sentence of true in is also true in . By Lemma 11 there is an elementary extension of such that embeds in . Then embeds in . Since and we have by Lemma 12 that . In particular, . But is squarelike so . It follows that . Hence, every universal sentence of true in is also true in . Since embeds in , every universal sentence of true in is also true in . Therefore, by Observation 3 of the previous section. That completes the induction and proves the lemma.

Proof of Theorem 9. (1) (2) Assume is squarelike. It will suffice to show that ucl() is axiomatizable by universal Horn sentences. Assume deducing a contradiction that is a universal sentence of true in but not a consequence of any set of universal Horn sentences of . Of course then itself cannot be a Horn sentence. We may assume that the matrix of is written in conjunctive normal form and hence has the form where each is a disjunction of literals. Thus, is equivalent to the conjunction of negated primitive sentences . Furthermore, at least one must contain at least two atomic disjuncts or else would be a Horn sentence. We claim that at least one of the negated primitive sentences containing at least two atomic disjuncts cannot be a consequence of any set of universal Horn sentences of true in . Suppose not. For each such that contains at least two atomic disjuncts let be a set of universal Horn sentences of true in such that is a consequence of . For each for which contains at most one atomic disjunct, is already a universal Horn sentence. For such we take and let . It follows then that would be a consequence of the set of universal Horn sentences of true in , contrary to hypothesis. The contradiction shows that at least one negated primitive sentence for which contains at least two atomic disjuncts cannot be a consequence of any set of universal Horn sentences of true in .
Fix such a conjunct (we suppress notationally). Let be where . For each fixed let be the quasi-identity Suppose deducing a contradiction that is true in . Then for every tuple from , would be true in . Equivalently, for every tuple from , would hold in . Since a disjunction is true if at least one of its disjuncts is true we would have in the above event that holds in for every tuple from . This implies that would be a consequence of the quasi-identity . But quasi-identities are special cases of universal Horn sentences and was postulated not to be a consequence of any set of universal Horn sentences of true in . The contradiction shows that none of the quasi-identities can hold in . Thus, for each , the existential sentence holds in .
Now let be a tuple of elements from such that simultaneously for all and . Let be the tuple from so that holds in . By Lemma 13, . It follows that the existential sentence holds in . But that contradicts the fact that its negation (up to logical equivalence) holds in . The contradiction shows that must have at least one set of universal Horn axioms. Hence, if is squarelike.
(2) (3) Suppose . Now and uhc() is closed under taking direct products. Let be an infinite index set and let . Then so is a model of and every universal sentence of true in is also true in . Thus, by Observation 3 of the previous section. Now is isomorphic to so that is discriminating by Theorem 6.
(3) (1) Suppose that where is discriminating. Then, in particular, every universal sentence of true in must also be true in . Thus, by Lemma 11, embeds in an elementary extension of . Now embeds in which is elementarily equivalent to by Lemma 12. In particular, . But is discriminating so by Theorem 7. Now so ultimately . Thus, every universal sentence of true in is also true in . But embeds in so every universal sentence of true in must also be true in . Then by Observation 3 of the previous section. That is, is squarelike.