Axiomatisability for topological quasivarieties: groups and semigroups
by
Marcel Jackson
La Trobe University

The quasivariety Q(A) generated by a finite algebra A is the class of all algebras that are sub-algebras of non-empty powers of A. The universal Horn theory Th_uH(A) of A is the set of formulas of the form

(p1 = q1 & p2 = q2 & ... & pn-1 = qn-1) --> pn = qn

or

~ (p1 = q1) \/ ~ (p2 = q2) \/ ... \/ ~ (pn = qn)

(for some n and some terms p_i, q_i) that are satisfied by A. The cancellative law for semigroups xy = zy --> x = z is an example of a formula of the first kind.

The topological quasivariety Q_T(A) of a finite algebra A is the class of algebras homeomorphic to a closed subalgebra of a power of A, where A, as a finite algebra, is given the discrete topology. Topologically, all such algebras have Boolean topology; that is, they must be compact, Hausdorff and have a basis of clopen sets.

A classical result of Mal'Cev states that Q(A) is exactly the class of all algebras satisfying the formulas in Th_uH(A). However, it is a strange fact that Mal'cev's theorem no longer holds for topological quasivarieties: while all members of Q_T(A) satisfy Th_uH(A), it is sometimes possible to find a Boolean topological algebra satisfying Th_uH(A) but not lying in Q_T(A). In this case we say that A is non-standard. We say that A is inherently non-standard if whenever A is in the quasivariety of some finite algebra B, then B is non-standard.

In this talk we give the first examples of inherently non-standard algebras and give a description of all such algebras in the class of finite groups and more generally, finite completely simple semigroups. For these examples, the property turns out to be related to several other well known properties such as residual character and finite axiomatisability.

Date received: August 4, 2003