Residual bounds, profiniteness and the axiomatisation of Boolean topological quasivarieties
by
Marcel Jackson
La Trobe University, Australia.
Coauthors: D. Clark, B. Davey, R. Freese and J. Pitkethly

An algebra is profinite if it is an inverse limit of finite (discretely topologised) algebras. Profinite algebras inherit a natural Boolean topology and are always (topologically) residually finite; conversely, every Boolean topological algebra that is topologically residually finite is profinite. Amongst semigroups, groups and rings, all Boolean topological algebras are residually finite (and therefore profinite). The first examples of Boolean topological algebras that are not topologically residually finite were given independently by W. Taylor and B. Banaschewski in the early 1970's.

In this talk we will examine a number of recent investigations relating to residual properties of Boolean topological algebras. We look at the problem of deciding when all Boolean topological models of a given finite set of identities are profinite and examine the underlying connections with properties such as the syntactic congruence from formal language theory, a natural generalisation of definable principal congruences, finite axiomatisability for quasivarieties, and the recently introduced notion of standardness for finite algebras.

Date received: June 16, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoc-05.