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


Authors: Dilip Raghavan and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 369 (2017), 4433-4455
MSC (2010): Primary 03E50, 03E05, 03E35, 54D80
DOI: https://doi.org/10.1090/tran/6943
Published electronically: January 9, 2017
MathSciNet review: 3624416
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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 ) \slash \textup {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.


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Additional Information

Dilip Raghavan
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076
Email: dilip.raghavan@protonmail.com

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/tran/6943
Keywords: Rudin-Keisler order, ultrafilter, P-point
Received by editor(s): November 9, 2014
Received by editor(s) in revised form: May 10, 2015, and January 28, 2016
Published electronically: January 9, 2017
Additional Notes: The first author was partially supported by National University of Singapore research grant number R-146-000-161-133.
The second author’s research was partially supported by European Research Council grant 338821. Publication 1058 on Shelah’s list.
Both authors’ research was partially supported by NSF grants DMS 0600940 and DMS 1101597.
Article copyright: © Copyright 2017 American Mathematical Society