Characterizing existence of certain ultrafilters

Abstract

Following Baumgartner (1995) [4], for an ideal I on ω, we say that an ultrafilter U on ω is an I-ultrafilter if for every function f:ω→ω there is A∈U with f[A]∈I.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:ω→ω there is A∈I with f−1[A]∉J). On the other hand, in general I≰KJ does not imply that existence of an I-ultrafilter which is not a J-ultrafilter is consistent.We provide some sufficient conditions on ideals to obtain the equivalence: I≰KJ if and only if it is consistent that there exists an I-ultrafilter which is not a J-ultrafilter. In some cases when the Katětov order is not enough for the above equivalence, we provide other conditions for which a similar equivalence holds. We are mainly interested in the cases when the family of all I-ultrafilters or J-ultrafilters coincides with some known family of ultrafilters: P-points, Q-points or selective ultrafilters (a.k.a. Ramsey ultrafilters). In particular, our results provide a characterization of Borel ideals I which can be used to characterize P-points as I-ultrafilters.Moreover, we introduce a cardinal invariant which is used to obtain a sufficient condition for the existence of an I-ultrafilter which is not a J-ultrafilter. Finally, we prove some new results concerning existence of certain ultrafilters under various set-theoretic assumptions.