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The study of the global structure of ultrafilters on the natural numbers withrespect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility wasinitiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paperBlass studied the special class of P-points under the quasi-ordering ofRudin-Keisler reducibility. He asked what partially ordered sets can beembedded into the P-points when the P-points are equipped with this ordering.This question is of most interest under some hypothesis that guarantees theexistence of many P-points, such as Martin's axiom for $\sigma$-centeredposets. In his 1973 paper he showed under this assumption that both${\omega}_{1}$ and the reals can be embedded. This result was later repeatedfor the coarser notion of Tukey reducibility. We prove in this paper thatMartin's axiom for $\sigma$-centered posets implies that every partial order ofsize at most continuum can be embedded into the P-points both underRudin-Keisler and Tukey reducibility.

On embedding certain partial orders into the P-points under RK and Tukey reducibility