THE RUDIN–KEISLER ORDERING OF P-POINTS UNDER 𝔟 = 𝔠

Abstract

AbstractM. E. Rudin (1971) proved, under CH, that for each P-point p there exists a P-point q strictly RK-greater than p. This result was proved under ${\mathfrak {p}= \mathfrak {c}}$ by A. Blass (1973), who also showed 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-ordering. In this paper, the results cited above are proved under the (weaker) assumption that $\mathfrak { b}=\mathfrak {c}$ . A. Blass asked in 1973 which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than $ \mathfrak {c}^{+}$ . In this paper it is proved, under $\mathfrak {b}=\mathfrak {c}$ , that for each ordinal $\alpha < \mathfrak {c}^{+}$ , there is an order embedding of $ \alpha $ into P-points. It is also proved, under $\mathfrak {b}=\mathfrak {c}$ , that there is an embedding of the long line into P-points.