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

by Raghavan and Shelah. [RaSh:1058]

Transactions American Math Soc, 2017

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.

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