## $G_\delta $ and co-meager semifilters

### Volume 235 / 2016

#### Abstract

The ultrafilters on the partial order $([\omega ]^{\omega },\subseteq ^*)$ are the free ultrafilters on $\omega $, which constitute the space $\omega ^*$, the Stone–Čech remainder of $\omega $. If $U$ is an upperset of this partial order (i.e., a *semifilter*), then ultrafilters on $U$ correspond to closed subsets of $\omega ^*$ 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 $[\omega ]^\omega $. 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 $G_\delta $ (and hence also co-meager) and $\mathfrak {d=c}$, 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 $(\omega ^*,+)$. This was an outstanding open problem solved only recently by Zelenyuk.