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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
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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  • [1] Tomek Bartoszyński and Haim Judah, Set theory: On the structure of the real line, A K Peters, Ltd., Wellesley, MA, 1995. MR 1350295
  • [2] J. Baumgartner, R. Frankiewicz, and P. Zbierski, Embedding of Boolean algebras in $ P(\omega )/{\rm fin}$, Fund. Math. 136 (1990), no. 3, 187-192. MR 1095691
  • [3] Andreas Blass, The Rudin-Keisler ordering of $ P$-points, Trans. Amer. Math. Soc. 179 (1973), 145-166. MR 0354350
  • [4] Andreas Blass, Near coherence of filters. I. Cofinal equivalence of models of arithmetic, Notre Dame J. Formal Logic 27 (1986), no. 4, 579-591. MR 867002,
  • [5] Andreas Blass, Near coherence of filters. II. Applications to operator ideals, the Stone-Čech remainder of a half-line, order ideals of sequences, and slenderness of groups, Trans. Amer. Math. Soc. 300 (1987), no. 2, 557-581. MR 876466,
  • [6] Andreas Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395-489. MR 2768685,
  • [7] Andreas Blass, Mauro Di Nasso, and Marco Forti, Quasi-selective ultrafilters and asymptotic numerosities, Adv. Math. 231 (2012), no. 3-4, 1462-1486. MR 2964612,
  • [8] Andreas Blass and Saharon Shelah, Near coherence of filters. III. A simplified consistency proof, Notre Dame J. Formal Logic 30 (1989), no. 4, 530-538. MR 1036674,
  • [9] W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267
  • [10] C. A. Di Prisco and S. Todorcevic, Souslin partitions of products of finite sets, Adv. Math. 176 (2003), no. 1, 145-173. MR 1978344,
  • [11] N. Dobrinen, J. Mijares, and T. Trujillo, Topological Ramsey spaces from Fraisse classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points, Arch. Math. Logic (to appear).
  • [12] Natasha Dobrinen and Stevo Todorcevic, Tukey types of ultrafilters, Illinois J. Math. 55 (2011), no. 3, 907-951 (2013). MR 3069290
  • [13] Natasha Dobrinen and Stevo Todorcevic, A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, Part 1, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1659-1684. MR 3145746,
  • [14] Natasha Dobrinen and Stevo Todorcevic, A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2, Trans. Amer. Math. Soc. 367 (2015), no. 7, 4627-4659. MR 3335396,
  • [15] Zdeněk Frolík, Sums of ultrafilters, Bull. Amer. Math. Soc. 73 (1967), 87-91. MR 0203676
  • [16] Kenneth Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299-306. MR 0314619
  • [17] Kenneth Kunen, Set theory: An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 597342
  • [18] K. Kunen, Weak $ P$-points in $ {\bf N}^{\ast } $, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 741-749. MR 588822
  • [19] Paul B. Larson, The filter dichotomy and medial limits, J. Math. Log. 9 (2009), no. 2, 159-165. MR 2679437,
  • [20] Jimena Llopis and Stevo Todorcevic, Borel partitions of products of finite sets, Acta Cient. Venezolana 47 (1996), no. 2, 85-88. MR 1453652
  • [21] David Milovich, Tukey classes of ultrafilters on $ \omega $, Spring Topology and Dynamics Conference, Topology Proc. 32 (2008), no. Spring, 351-362. MR 1500094
  • [22] David Milovich, Forbidden rectangles in compacta, Topology Appl. 159 (2012), no. 14, 3180-3189. MR 2948276,
  • [23] Dilip Raghavan and Stevo Todorcevic, Cofinal types of ultrafilters, Ann. Pure Appl. Logic 163 (2012), no. 3, 185-199. MR 2871264,
  • [24] Andrzej Rosłanowski and Saharon Shelah, Norms on possibilities. I. Forcing with trees and creatures, Mem. Amer. Math. Soc. 141 (1999), no. 671, xii+167. MR 1613600,
  • [25] Mary Ellen Rudin, Partial orders on the types in $ \beta N$, Trans. Amer. Math. Soc. 155 (1971), 353-362. MR 0273581
  • [26] Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409-419. MR 0080902
  • [27] Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206
  • [28] Saharon Shelah and Jindřich Zapletal, Ramsey theorems for product of finite sets with submeasures, Combinatorica 31 (2011), no. 2, 225-244. MR 2848252,
  • [29] Stevo Todorcevic, Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010. MR 2603812

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Additional Information

Dilip Raghavan
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076

Saharon Shelah
Affiliation: Einstein Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel

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

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