Atlas home || Conferences | Abstracts | about Atlas

Algebras, Lattices, Varieties - A Conference in Honor of Walter Taylor
August 15-18, 2004
University of Colorado
Boulder, Colorado, USA

Organizers
Jennifer Hyndman, Keith Kearnes, Ralph McKenzie, George McNulty, Ágnes Szendrei, Ross Willard

View Abstracts
Conference Homepage

Profinite structures and the axiomatisability of topological quasi-varieties
by
Brian Davey
La Trobe University (Australia)
Coauthors: David Clark, Marcel Jackson, Jane Pitkethly

The paper on which this talk is based grew out of an effort to decide when the topological quasi-varieties that arise in the theory of natural dualities can be described axiomatically. The resulting research effort has taken on a life of its own quite independent of duality theory. In this talk we restrict our attention to structures that have (finitary) operations and relations but no partial operations.

Define a class K of finite (discretely topologised) structures to be standard if the topological quasi-variety IScP+(K) generated by K is precisely the Boolean topological models of the universal Horn theory of K. (Thus, when K is finite, the topological quasi-variety IScP+(K) is standard if the standard universal Horn description of ISP+(K) also serves to describe IScP+(K).)
To illustrate the sorts of results we have obtained, I shall present
As sample applications of (1) we can show that the classes of all finite graphs and of all finite di-graphs are standard.
Nesetril and Pultr have shown that there is an (n+2)-element graph Ck such that ISP+({Ck}) is precisely the class of k-colourable graphs. We prove that a Boolean topological simple graph is topologically k-colourable (= clopen k-partite) and contained in IScP+({Ck}), for some k, from which it follows that the class consisting of all finite simple graphs is standard.
By way of contrast, we apply (2) above to show that the graph Ck itself is not standard since there is a Boolean topological simple graph that is k-colourable as a graph but is not topologically k-colourable. This also shows that the class of k-colourable graphs has no finite basis for its universal Horn theory, a result first proved by Walter Taylor in 1969.

Date received: July 8, 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-19.