Configurations in coproducts of Priestley spaces
by
Richard N. Ball
Department of Mathematics, University of Denver, Denver Colorado 80208, U.S.A.
Coauthors: Ales Pultr

A coproduct of Priestley spaces (compact Hausdorff "order-zero-dimensional" ordered spaces) is a suitable compactification of the disjoint union, not yet fully understood. In particular there is the natural question of which configurations P (finite connected posets) cannot occur in a coproduct without appearing first in some of the summands. This problem has been recently completely answered by the authors and J. Sichler for configurations possessing a top element: in this case these are precisely the trees. In the more general case one can prove that combinatorial trees (configurations the Hasse diagrams of which, considered as symmetric graphs, are trees) have the property. Because the variety of combinatorial trees and their structure considerably exceeds that of the rooted trees, this fact is more complex. It is conjectured that, analogous to the topped case, there will generally be no other configurations with this property. This surmise is strongly supported by some negative results concerning configurations with cycles. There are, however, still open problems that seem to be hard.

In the talk we will briefly explain the situation with top element, present some details on the positive part of the general case, present some of the negative results, and explain the dificulties with some special cases. Furthermore, some more details on the shape of the coproduct will be presented, and the relation of the facts above with the axiomatizability of classes of distributive lattices obtained by prohibiting configurations in their Priestley duals will be discussed.

Date received: July 10, 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-28.