Standard topological quasi-varieties and syntactic congruences revisited
by
Brian A. Davey
La Trobe University
Coauthors: David M. Clark (New Paltz), Ralph S. Freese (Hawaii), Marcel Jackson (La Trobe)

In this talk we present some useful observations linking syntactic congruences in the quasi-variety generated by a finite algebra M = <M;G> and the axiomatisation of the topological quasi-variety generated by the topological algebra M_T = <M;G, T> obtained by adding the discrete topology T to M. We begin by considering a more general situation which serves as part of the abstract of this and the following two talks.

Let M = <M;G, R> be a finite structure. A famous theorem of Mal'cev states that the class ISP⁺(M) consisting of all Isomorphic copies of Substructures of non-zero Powers of M is equal to the class Mod(F) consisting of all models of a family F of universally quantified Horn formulæ. (A Horn formula in the first-order language of M is an expression of the form u or ( ~ v₁ \/ ... \/ ~ v_n) or ( v₁ /\ ... /\ v_n) ===> u, where u and each v_i are atomic formulæ.)

If we consider the discretely topologised structure M_T = <M;G, R, T> and restrict our attention to the topological quasi-variety Q_T(M) : = IS_cP⁺(M_T) consisting of all Isomorphic copies of topologically closed Substructures of non-zero Powers of M_T, the situation becomes much more complicated and mysterious. Assume that ISP⁺(M) = Mod(F), á la Mal'cev. Then every structure X = <X; G, R, T > in Q_T(M) is a Boolean topological model of F, i.e. <X; T> is a Boolean topological space, each operation g in G is continuous, each (n-ary) relation r in R is closed (in Xⁿ) and <X; G, R> is a model of F. If, conversely, every Boolean topological model of F is in Q_T(M), then the structure M_T is called standard and Q_T(M) is referred to as a standard topological quasi-variety. For example, a finite cyclic group or Boolean algebra (with the discrete topology added) is standard while the two element chain as an ordered set (with the discrete topology added) is non-standard.

In this talk we restrict ourselves to the case where M is an algebra. Drawing on the formal language notion of syntactic congruences, we prove that M_T is standard provided that the algebraic quasi-variety Q(M):=ISP(M) generated by M is a variety, and that the class Q(M) has finitely determined syntactic congruences (FDSC), i.e. syntactic congruences on the algebras in Q(M) are determined by a finite set of terms. This gives purely algebraic conditions that yield a topological conclusion.

We also prove if X is a compact topological algebra which has FDSC, then X is profinite (i.e. X is isomorphic to an inverse limit of finite algebras) iff X is topologically residually finite iff X is a Boolean topological algebra. For semigroups and groups (which are known to have FDSC) this result is well known but it does not seem to appear in the literature in its general form. It provides a topological condition which yields the purely algebraic conclusion: if A is a variety and we can find a Boolean topological A-algebra that is not topologically residually finite, then A does not have FDSC. Moreover, if ISP⁺(M) = Mod(F) and we can find a Boolean topological model of F that is not topologically residually finite, then not only is M_T non-standard, it is inherently non-standard, that is, any finitely generated topological quasi-variety that contains M_T is non-standard. Several applications of these ideas will be presented.

Date received: September 24, 2003