# Ultrafilters, monotone functions and pseudocompactness

Archive for Mathematical Logic 44 (2):131-157 (2005)

 Authors Abstract In this article we, given a free ultrafilter p on ω, consider the following classes of ultrafilters:(1) T(p) - the set of ultrafilters Rudin-Keisler equivalent to p,(2) S(p)={q ∈ ω*:∃ f ∈ ω ω , strictly increasing, such that q=f β (p)},(3) I(p) - the set of strong Rudin-Blass predecessors of p,(4) R(p) - the set of ultrafilters equivalent to p in the strong Rudin-Blass order,(5) P RB (p) - the set of Rudin-Blass predecessors of p, and(6) P RK (p) - the set of Rudin-Keisler predecessors of p,and analyze relationships between them. We introduce the semi-P-points as those ultrafilters p ∈ ω* for which P RB (p)=P RK (p), and investigate their relations with P-points, weak-P-points and Q-points. In particular, we prove that for every semi-P-point p its α-th left power α p is a semi-P-point, and we prove that non-semi-P-points exist in ZFC. Further, we define an order ⊴ in T(p) by r⊴q if and only if r ∈ S(q). We prove that (S(p),⊴) is always downwards directed, (R(p),⊴) is always downwards and upwards directed, and (T(p),⊴) is linear if and only if p is selective.We also characterize rapid ultrafilters as those ultrafilters p ∈ ω* for which R(p)∖S(p) is a dense subset of ω*.A space X is M-pseudocompact (for ) if for every sequence (U n ) n < ω of disjoint open subsets of X, there are q ∈ M and x ∈ X such that x=q-lim (U n ); that is, for every neighborhood V of x. The P RK (p)-pseudocompact spaces were studied in [ST].In this article we analyze M-pseudocompactness when M is one of the classes S(p), R(p), T(p), I(p), P RB (p) and P RK (p). We prove that every Frolik space is S(p)-pseudocompact for every p ∈ ω*, and determine when a subspace with is M-pseudocompact Keywords Rudin-Keisler pre-order  Rudin-Blass pre-order   M-pseudocompactness  Semi-P-points  Rapid filters  P-points  Q-points  Selective ultrafilters Categories (categorize this paper) Reprint years 2005 DOI 10.1007/s00153-004-0246-y Options Mark as duplicate Export citation Request removal from index

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## References found in this work BETA

Ultrafilters on a Countable Set.David Booth - 1970 - Annals of Mathematical Logic 2 (1):1.
The Rudin-Blass Ordering of Ultrafilters.Claude Laflamme & Jian-Ping Zhu - 1998 - Journal of Symbolic Logic 63 (2):584-592.

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