On strong \(P\)-points
type: Item
instance of: scholarly article
zbMATH Open document ID: 1302.03050
DOI: 10.1090/S0002-9939-2013-11518-2
author: Andreas Blass
author: Michael Hrušák
author: Jonathan Verner
published in: Proceedings of the American Mathematical Society
publication date: 2013-07-05T00:00:00Z
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 6184487
zbMATH Keywords: ultrafilter
zbMATH Keywords: Mathias forcing
zbMATH Keywords: strong \(P\)-point
MaRDI profile type: Publication
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full work available at URL: s0002-9939-2013-11518-2
OpenAlex ID: W1858634293
title: On strong \(P\)-points
review text: \textit{M. Canjar} [Ann. Pure Appl. Logic 37, No. 1, 1--79 (1988; Zbl 0646.03025)] first investigated the question: When does the Mathias forcing relativized to an ultrafilter on \(\omega\) add no dominating reals (it is called \textit{Canjar ultrafilter} in this paper)? Later, \textit{C. Laflamme} [Ann. Pure Appl. Logic 42, No. 2, 125--163 (1989; Zbl 0681.03035)] introduced the notion of a \textit{strong \(P\)-point} and noted it as a necessary condition for a Canjar ultrafilter. \textit{M. Hrušák} and \textit{H. Minami} recently introduced [Ann. Pure Appl. Logic 165, No. 3, 880--894 (2014; Zbl 1306.03023)] an equivalent combinatorial condition for being a Canjar ultrafilter. Via this characterization, the authors show in this paper that an ultrafilter on \(\omega\) is a Canjar ultrafilter iff it is a strong \(P\)-point.NEWLINENEWLINECanjar showed in [loc. cit.] that a Canjar ultrafilter must be a \(P\)-point with no rapid Rudin-Keisler predecessor. In Section 3 of this paper, under the hypothesis that cov\(({\mathcal M})={\mathfrak c}\), the authors construct a \(P\)-point which is not strong, and neither has a rapid Rudin-Keisler predecessor.
reviewed by: Xianghui Shi
description: scientific article; zbMATH DE number 6184487