Abstract
The ultrafilters on the partial order ([ω]ω , SUBSET OF OR EQUAL TO ASTERISK OPERATOR ) are the free ultrafilters on ω, which constitute the space ω ASTERISK OPERATOR , the Stone-Cech remainder of ω. If U is an upperset of this partial order (i.e., a semifilter ), then the ultrafilters on U correspond to closed subsets of ω ASTERISK OPERATOR via Stone duality.
If U is large enough, then it is possible to get combinatorially nice ultrafilters on U by generalizing the corresponding constructions for [ω]ω. In particular, if U is co-meager then there are ultrafilters on U that are weak P-filters (extending a result of Kunen).
If U is Gδ (and hence also co-meager) and d = c then there are ultrafilters on U that are P-filters (extending a result of Ketonen). For certain choices of U , these constructions have applications in dynamics, algebra, and combinatorics.
Most notably, we give a new proof of the fact that there are minimal-maximal idempotents in (ω*, +). This was an outstanding open problem solved only last year by Zelenyuk.