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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On strong $ P$-points


Authors: Andreas Blass, Michael Hrušák and Jonathan Verner
Journal: Proc. Amer. Math. Soc. 141 (2013), 2875-2883
MSC (2010): Primary 03E05, 03E17, 03E35
Published electronically: April 3, 2013
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Abstract: This paper investigates the combinatorial property of ultrafilters where the Mathias forcing relativized to them does not add dominating reals. We prove that the characterization due to Hrušák and Minami is equivalent to the strong $ P$-point property. We also consistently construct a $ P$-point that has no rapid Rudin-Keisler predecessor but that is not a strong $ P$-point. These results answer questions of Canjar and Laflamme.


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Additional Information

Andreas Blass
Affiliation: Mathematics Department, University of Michigan, Ann Arbor, Michigan 48109–1043
Email: ablass@umich.edu

Michael Hrušák
Affiliation: Instituto de Matematicas, UNAM, Apartado postal 61-3, Xangari, 58089, Morelia, Michoacin, Mexico
Email: michael@matmor.unam.mx

Jonathan Verner
Affiliation: Department of Logic, Charles University, Palachovo nám. 2, 116 38 Praha 1, Czech Republic
Email: jonathan.verner@matfyz.cz

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11518-2
PII: S 0002-9939(2013)11518-2
Keywords: Ultrafilter, Mathias forcing, strong $P$-point
Received by editor(s): February 23, 2011
Received by editor(s) in revised form: October 26, 2011
Published electronically: April 3, 2013
Additional Notes: The first author was partially supported by NSF grant DMS-0653696.
The second author was partially supported by PAPIIT grant IN101608 and CONACyT grant 80355.
The third author was partially supported by GAR 401/09/H007 Logické základy sémantiky
Communicated by: Julia Knight
Article copyright: © Copyright 2013 American Mathematical Society