Coherent ultrafilters and nonhomogeneity

TL;DR

This paper introduces coherent ultrafilters on Boolean algebras, demonstrating their generic existence under certain set-theoretic assumptions and applying them to show nonhomogeneity in topological spaces.

Contribution

It extends the concept of $P$-points to coherent ultrafilters on Boolean algebras and proves their generic existence under specific cardinal equalities.

Findings

Existence of coherent $P$-ultrafilters under ${rak c}={rak d}$

Existence of coherently selective ultrafilters under ${rak c}={cov(M)}$

Application to nonhomogeneity in Stone spaces

Abstract

We introduce the notion of a coherent $P$-ultrafilter on a complete ccc Boolean algebra, strenghtening the notion of a $P$-point on $\omega$, and show that these ultrafilters exist generically under ${\mathfrak c} = {\mathfrak d}$. This improves the known existence result of Ketonen. Similarly, the existence theorem of Canjar can be extended to show that coherently selective ultrafilters exist generically under ${\mathfrak c} = {cov(M)}$. We use these ultrafilters in a topological application: a coherent $P$-ultrafilter on an algebra $B$ is an untouchable point in the Stone space of $B$, witnessing its nonhomogeneity.