Abstract:We introduce the notion of a {\it coherent $P$-ultrafilter\/} on a complete ccc Boolean algebra, strengthening 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 [{\it On the existence of $P$-points in the Stone-\v Cech compactification of integers\/}, Fund. Math. {\bf 92} (1976), 91--94]. Similarly, the existence theorem of Canjar [{\it On the generic existence of special ultrafilters\/}, Proc. Amer. Math. Soc. {\bf 110} (1990), no.~1, 233--241] can be extended to show that {\it coherently selective ultrafilters\/} exist generically under $\mathfrak c = \operatorname{cov}\mathcal M$. We use these ultrafilters in a topological application: a coherent $P$-ultrafilter on an algebra $\mathcal B$ is an {\it untouchable point\/} in the Stone space of $\mathcal B$, witnessing its nonhomogeneity.
Keywords: nonhomogeneity; ultrafilter; Boolean algebra; untouchable point
DOI: DOI 10.14712/1213-7243.2015.123
AMS Subject Classification: 54G05 06E10