$G_\delta $ and co-meager semifilters

Volume 235 / 2016

William R. Brian, Jonathan L. Verner Fundamenta Mathematicae 235 (2016), 153-166 MSC: Primary 54D35; Secondary 22A15, 03E35, 06A07. DOI: 10.4064/fm182-2-2016 Published online: 30 May 2016


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.


  • William R. BrianDepartment of Mathematics
    Tulane University
    6823 St. Charles Ave.
    New Orleans, LA 70118, U.S.A.
  • Jonathan L. VernerDepartment of Logic
    Faculty of Arts Charles University Palachovo nám. 2
    116 38 Praha 1, Czech Republic

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