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

Abstract:

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for $\sigma$-centered posets. In his 1973 paper he showed under this assumption that both ${\omega }_{1}$ and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for $\sigma$-centered posets implies that the Boolean algebra $\mathcal {P}(\omega ) / \operatorname {FIN}$ equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.