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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

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