Coherent ultrafilters and nonhomogeneity

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.