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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.



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[1]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. 7990.
[2]Blass, Andreas, Selective ultrafilters and homogeneity. Annals of Pure and Applied Logic, vol. 38 (1988), pp. 215255.
[3]Di Prisco, Carlos and Todorcevic, Stevo, Souslin partitions of products of finite sets. Advances in Mathematics, vol. 176 (2003), pp. 145173.
[4]Dobrinen, Natasha, Continuous and other finitely generated canonical cofinal maps on ultrafilters, 38 pp., submitted. arXiv:1505.00368
[5]Dobrinen, Natasha and Todorcevic, Stevo, Tukey types of ultrafilters. Illinois Journal of Mathematics, vol. 55 (2011), no. 3, pp. 907951.
[6]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. 16591684.
[7]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. 46274659.
[8]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.
[9]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. 2733.
[10]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. 741749.
[11]Mazur, Krzysztof, Fσ-ideals and $\omega _1 \omega _1^{\rm{*}}$-gaps in the Boolean algebras P(ω)/I. Fundamenta Mathematicae, vol. 138 (1991), pp. 103111.
[12]Mijares, José G. and Nieto, Jesús, Local Ramsey theory. An abstract approach, arXiv:0712.2393v1 2013, pp. 11.
[13]Raghavan, Dilip, The generic ultrafilter added by (Fin × Fin)+, 2012, p. 7, arXiv:1210.7387.
[14]Raghavan, Dilip and Todorcevic, Stevo, Cofinal types of ultrafilters. Annals of Pure and Applied Logic, vol. 163 (2012), pp. 185199.
[15]Solecki, Sławomir and Todorcevic, Stevo, Cofinal types of topological directed orders. Annales de l’institut Fourier (Grenoble), vol. 54 (2004), pp. 18771911.
[16]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. 577584.
[17]Todorcevic, Stevo, Gaps in analytic quotients. Fundamenta Mathematicae, vol. 156 (1998), pp. 8597.
[18]The Online Encyclopedia of Integer Sequences,



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