Standard Topological Algebras and Syntactic Congruences
by
David M. Clark
SUNY New Paltz
Coauthors: Brian Davey, Miroslav Haviar, Marcel Jackson, Jane Pitkethly, Rashed Talukder

A topological quasi-variety (TQV) X is a category obtained from a finite algebraic structure M carrying the discrete topology by closing {M} under the formation of direct products, topologically closed substructures and isomorphic images. TQVs are of interest to algebraists since they arise as the duals of algebraic quasi-varieties under natural dualities. In order to make use of a natural duality, it is necessary to have a clear understanding of the structure of the members of its dual category X. A standard topological quasi-variety (STQV) is a TQV in which such an understanding arises in a canonical fashion: X consists exactly of those algebraic structures having the type of M which carry a compatible Boolean topology and are models of the quasi-atomic theory of M. We present a congruence condition based on the terms of a topological algebra M which guarantees that the TQV it generates is standard.

Date received: January 4, 2002