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THE NEXT BEST THING TO A P-POINT

Published online by Cambridge University Press:  22 July 2015

ANDREAS BLASS ,
NATASHA DOBRINEN  and
DILIP RAGHAVAN
ANDREAS BLASS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MICHIGAN ANN ARBOR, MI 48109–1043 USAE-mail: ablass@umich.eduURL: http://www.math.lsa.umich.edu/∼ablass
NATASHA DOBRINEN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF DENVER 2360 GAYLORD ST DENVER, CO 80208 USAE-mail: natasha.dobrinen@du.eduURL: http://web.cs.du.edu/∼ndobrine
DILIP RAGHAVAN
Affiliation:
DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 119076E-mail: dilip.raghavan@protonmail.comURL: http://www.math.toronto.edu/∼raghavan
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Abstract

We study ultrafilters on ω2 produced by forcing with the quotient of ${\cal P}$(ω2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.

Keywords

P-pointselective ultrafilteranalytic quotientweak partition relationTukey orderRudin-Keisler ordertensor product
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 3 , September 2015 , pp. 866 - 900
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

REFERENCES

Blass, Andreas, A model-theoretic view of some special ultrafilters, Logic Colloquium ’77 (Macintyre, A., Pacholski, L., and Paris, J., editors), Studies in Logic and the Foundations of Mathematics, vol. 96, North-Holland, Amsterdam, 1978, pp. 79–90.CrossRefGoogle Scholar
Blass, Andreas, Selective ultrafilters and homogeneity. Annals of Pure and Applied Logic, vol. 38 (1988), pp. 215–255.CrossRefGoogle Scholar
Di Prisco, Carlos and Todorcevic, Stevo, Souslin partitions of products of finite sets. Advances in Mathematics, vol. 176 (2003), pp. 145–173.CrossRefGoogle Scholar
Dobrinen, Natasha, Continuous and other finitely generated canonical cofinal maps on ultrafilters, 38 pp., submitted. arXiv:1505.00368Google Scholar
Dobrinen, Natasha and Todorcevic, Stevo, Tukey types of ultrafilters. Illinois Journal of Mathematics, vol. 55 (2011), no. 3, pp. 907–951.CrossRefGoogle Scholar
Dobrinen, Natasha, A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 1. Transactions of American Mathematical Society, vol. 366 (2014), no. 3, pp. 1659–1684.CrossRefGoogle Scholar
Dobrinen, Natasha, A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2. Transactions of the American Mathematical Society, vol. 367 (2015), no. 7, pp. 4627–4659.CrossRefGoogle Scholar
Farah, Ilijas, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Memoirs of the American Mathematical Society, vol. 148 (2000), no. 702, xvi+177 pp.CrossRefGoogle Scholar
Hernández-Hernández, Fernando, Distributivity of quotients of countable products of Boolean algebras. Rendiconti dell’Istituto di Matematica dell’Università di Trieste, vol. 41 (2009), pp. 27–33.Google Scholar
Kunen, Kenneth, Weak P-points in N*, Topology, Vol. II (Proceedings of Fourth Colloquium, Budapest, 1978, Colloquia of Mathematical Society, János Bolyai), vol. 23, North-Holland, 1980, pp. 741–749.Google Scholar
Mazur, Krzysztof, Fσ-ideals and $\omega _1 \omega _1^{\rm{*}}$-gaps in the Boolean algebras P(ω)/I. Fundamenta Mathematicae, vol. 138 (1991), pp. 103–111.CrossRefGoogle Scholar
Mijares, José G. and Nieto, Jesús, Local Ramsey theory. An abstract approach, arXiv:0712.2393v1 2013, pp. 11.Google Scholar
Raghavan, Dilip, The generic ultrafilter added by (Fin × Fin)+, 2012, p. 7, arXiv:1210.7387.Google Scholar
Raghavan, Dilip and Todorcevic, Stevo, Cofinal types of ultrafilters. Annals of Pure and Applied Logic, vol. 163 (2012), pp. 185–199.CrossRefGoogle Scholar
Solecki, Sławomir and Todorcevic, Stevo, Cofinal types of topological directed orders. Annales de l’institut Fourier (Grenoble), vol. 54 (2004), pp. 1877–1911.CrossRefGoogle Scholar
Szymański, Andrzej and Xua Zhou, Hao, The behavior of ω 2*under some consequences of Martin’s axiom, General Topology and Its Relations to Modern Analysis and Algebra. V, (Prague, 1981) (Novák, J., editor), Sigma Series in Pure Mathematics, vol. 3, Heldermann Verlag, 1983, pp. 577–584.Google Scholar
Todorcevic, Stevo, Gaps in analytic quotients. Fundamenta Mathematicae, vol. 156 (1998), pp. 85–97.CrossRefGoogle Scholar
The Online Encyclopedia of Integer Sequences, http://oeis.org.Google Scholar