Ultrafilters, monotone functions and pseudocompactness

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, 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) PRB(p) the set of Rudin-Blass predecessors of p, and (6) PRK(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 PRB(p) = PRK(p), and investigate their relations with P -points, weakP -points and Q-points. In particular, we prove that for every semi-P -point p its α-th left power 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 M ⊂ ω∗) if for every sequence (Un)n<ω of disjoint open subsets of X, there are q ∈ M and x ∈ X such that x = q-lim(Un); that is, {n < ω : V ∩ Un = ∅} ∈ q for every neighborhood V of x. The PRK(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), PRB(p) and PRK(p). We prove that every Frolı́k space is S(p)-pseudocompact for every p ∈ ω∗, and determine when a subspace X ⊂ βω with ω ⊂ X is M-pseudocompact. M. Hrušák: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Xangari, 58089 Morelia Michoacan, México. e-mail: [email protected] M. Sanchis: Departament de Matemàtiques, Universitat Jaume I, Campus de Riu Sec s/n, Castelló, Spain. e-mail: [email protected] Á.Tamariz-Mascarúa: Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, 04510, México. e-mail: [email protected] The first author’s research was partially supported by a grant GAČR 201/00/1466 Mathematics Subject Classification (2000): 54D80, 03E05, 54A20, 54D20