Ultrafilters and the Katětov order

Abstract

Let $I$ be an ideal on ω. Following Baumgartner (1995) [2], we say that an ultrafilter $U$ on ω is an $I$-ultrafilter if for every function $f:\omega \to \omega$ there is $A\in U$ with $f\left[A\right]\in I$. In particular, P-points are exactly $\mathrm{Fin}\times \mathrm{Fin}$-ultrafilters.

If there is an $I$-ultrafilter which is not a $J$-ultrafilter, then $I$ is not below $J$ in the Katětov order ${⩽}_K$ (i.e., for every function $f:\omega \to \omega$ there is $A\in I$ with $f^{-1}\left[A\right]\notin J$), however the reversed implication is not true (even consistently).

Recently it was shown that for all Borel ideals $I$ we have: $I{\nleqq }_K\mathrm{Fin}\times \mathrm{Fin}$ if and only if in some forcing extension one can find an $I$-ultrafilter which is not a P-point (Filipów et al. (2022) [6]).

We show that under some combinatorial assumptions imposed on the ideal $J$, the classes of $J$-ultrafilters and $\mathrm{Fin}\times J$-ultrafilters coincide. This allows us to find some sufficient conditions on ideals to obtain the equivalence: $I{\nleqq }_K\mathrm{Fin}\times J$ if and only if in some forcing extension one can find an $I$-ultrafilter which is not a $J$-ultrafilter. We provide several examples of ideals, for which the above equivalence is true, including the ideal of nowhere dense subsets of $Q$ and the ideal of sets of asymptotic density zero.