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Keywords:
nonhomogeneity; ultrafilter; Boolean algebra; untouchable point
Summary:
We introduce the notion of a {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 [{On the existence of $P$-points in the Stone-Čech compactification of integers}, Fund. Math. {92} (1976), 91--94]. Similarly, the existence theorem of Canjar [{On the generic existence of special ultrafilters}, Proc. Amer. Math. Soc. {110} (1990), no. 1, 233--241] can be extended to show that {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 {untouchable point} in the Stone space of $\mathcal B$, witnessing its nonhomogeneity.
References:
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