The Rudin-Kisler ordering of P-points under $\mathfrak{b} = \mathfrak{c}$

TL;DR

This paper extends previous results on the Rudin-Kisler ordering of P-points by proving key properties under the assumption that the bounding number equals the continuum, and constructs an embedding of + into P-points.

Contribution

It proves that the Rudin-Kisler order properties of P-points hold under = and constructs an embedding of + into P-points, answering Blass's question.

Findings

Established P-point properties under =.

Constructed an order embedding of + into P-points.

Extended Rudin's and Blass's results beyond CH.

Abstract

M. E. Rudin proved under CH that for each P-point there exists another P-point strictly RK-greater (M. E. Rudin, Partial orders on the types of $\beta \mathbb{N}$ , Trans. Amer. Math. Soc., 155 (1971), 353-362). Assuming $\mathfrak{p}=\mathfrak{c}$ A. Blass showed the same, and proved that each RK-increasing $\omega$-sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-(pre)ordering (A. Blass, Rudin - Keisler ordering on P-points, Trans. Amer. Math. Soc., 179 (1973), 145-166). In the present paper the results cited above are proved under a (weaker) assumption $\mathfrak{b}=\mathfrak{c}$. A. Blass also asked in (A. Blass, Rudin - Keisler ordering on P-points, Trans. Amer. Math. Soc., 179 (1973), 145-166) what ordinals can be embedded in the set of P-points and pointed out, that such an ordinal may not be greater then $\mathfrak{c}^+$. In the present paper the question is answered showing (under $\mathfrak{b} = \mathfrak{c}$) that there is an order embedding of $\mathfrak{c}^+$ into P-points.