Rudin-Kisler ordering on the P-hierarchy

TL;DR

This paper advances the understanding of the Rudin-Kisler ordering on P-points by extending results to higher classes of the P-hierarchy, assuming certain set-theoretic conditions, and demonstrating the existence of many incomparable upper bounds.

Contribution

It generalizes Blass's results to all classes of the P-hierarchy with assumptions weaker than MA, and shows the existence of numerous incomparable upper bounds.

Findings

Extended Rudin-Kisler ordering results to P-hierarchy classes ≥ 2.

Under $rak{b}=rak{c}$, each increasing sequence of P-points has many upper bounds.

Existence of at least $rak{b}$ many incomparable upper bounds for these sequences.

Abstract

M. E. Rudin in "Partial orders on the types of $\beta {N}$", (Trans. Amer. Math. Soc., 155, 1971, 353-362) proved (under CH) that for each P-point $u$ there is a P-point $v$ such that $v>_{RK}u$. A. Blass in "Rudin - Kisler ordering on P-points" (Trans. Amer. Math. Soc. 179, 1973, 145-166) improved that theorem assuming MA in the place of CH, in that paper he also proved that under MA each RK-increasing sequence of P-points is upper bounded by a P-point. We improve Blass results simultaneously in 3 directions - we prove it for each class of index $\geq 2$ of P-hierarchy (P-points coincidence with a class $P_2$ of P-hierarchy), assuming $\mathfrak{b}=\mathfrak{c}$ in the place of MA and we show that there are at least $\mathfrak{b}$ many Rudin-Kisler incomparable such upper bounds.