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

TL;DR

This paper demonstrates that under Martin's axiom for σ-centered posets, any partial order of size at most continuum can be embedded into the P-points with respect to Rudin-Keisler and Tukey reducibility, extending previous results.

Contribution

It proves that all partial orders of size at most continuum can be embedded into P-points under both reducibility notions assuming Martin's axiom.

Findings

Embedding of all partial orders of size continuum into P-points under MA(σ-centered).

Extension of previous embeddings of ω₁ and reals to all partial orders of size continuum.

Results hold for both 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. This result was later repeated for the coarser notion of Tukey reducibility. We prove in this paper that Martin's axiom for $\sigma$-centered posets implies that every partial order of size at most continuum can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility.