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

Dilip Raghavan, Saharon Shelah
2017 Transactions of the American Mathematical Society  
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 quasiordering 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
more » ... pothesis that guarantees the existence of many P-points, such as Martin's axiom for σ-centered posets. In his 1973 paper he showed under this assumption that both ω 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 σ-centered posets implies that the Boolean algebra P(ω)/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.
doi:10.1090/tran/6943 fatcat:m3esbkerq5azrg26trtyty5cpi