Archive for Mathematical Logic 44 (2):131-157 (2005)

Authors
Abstract
In this article we, given a free ultrafilter p on ω, consider the following classes of ultrafilters:(1) T(p) - the set of ultrafilters Rudin-Keisler equivalent to p,(2) S(p)={q ∈ ω*:∃ f ∈ ω ω , strictly increasing, such that q=f β (p)},(3) I(p) - the set of strong Rudin-Blass predecessors of p,(4) R(p) - the set of ultrafilters equivalent to p in the strong Rudin-Blass order,(5) P RB (p) - the set of Rudin-Blass predecessors of p, and(6) P RK (p) - the set of Rudin-Keisler predecessors of p,and analyze relationships between them. We introduce the semi-P-points as those ultrafilters p ∈ ω* for which P RB (p)=P RK (p), and investigate their relations with P-points, weak-P-points and Q-points. In particular, we prove that for every semi-P-point p its α-th left power α p is a semi-P-point, and we prove that non-semi-P-points exist in ZFC. Further, we define an order ⊴ in T(p) by r⊴q if and only if r ∈ S(q). We prove that (S(p),⊴) is always downwards directed, (R(p),⊴) is always downwards and upwards directed, and (T(p),⊴) is linear if and only if p is selective.We also characterize rapid ultrafilters as those ultrafilters p ∈ ω* for which R(p)∖S(p) is a dense subset of ω*.A space X is M-pseudocompact (for ) if for every sequence (U n ) n < ω of disjoint open subsets of X, there are q ∈ M and x ∈ X such that x=q-lim (U n ); that is, for every neighborhood V of x. The P RK (p)-pseudocompact spaces were studied in [ST].In this article we analyze M-pseudocompactness when M is one of the classes S(p), R(p), T(p), I(p), P RB (p) and P RK (p). We prove that every Frolik space is S(p)-pseudocompact for every p ∈ ω*, and determine when a subspace with is M-pseudocompact
Keywords Rudin-Keisler pre-order  Rudin-Blass pre-order   M-pseudocompactness  Semi-P-points  Rapid filters  P-points  Q-points  Selective ultrafilters
Categories (categorize this paper)
Reprint years 2005
DOI 10.1007/s00153-004-0246-y
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 61,089
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

Ultrafilters on a Countable Set.David Booth - 1970 - Annals of Mathematical Logic 2 (1):1.
The Rudin-Blass Ordering of Ultrafilters.Claude Laflamme & Jian-Ping Zhu - 1998 - Journal of Symbolic Logic 63 (2):584-592.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Some Initial Segments of the Rudin-Keisler Ordering.Andreas Blass - 1981 - Journal of Symbolic Logic 46 (1):147-157.
The Rudin-Blass Ordering of Ultrafilters.Claude Laflamme & Jian-Ping Zhu - 1998 - Journal of Symbolic Logic 63 (2):584-592.
Ultrafilters on Ω.James E. Baumgartner - 1995 - Journal of Symbolic Logic 60 (2):624-639.
Ultrafilters on $Omega$.James E. Baumgartner - 1995 - Journal of Symbolic Logic 60 (2):624-639.
P-Hierarchy on Β Ω.Andrzej Starosolski - 2008 - Journal of Symbolic Logic 73 (4):1202-1214.
Standardization Principle of Nonstandard Universes.Masahiko Murakami - 1999 - Journal of Symbolic Logic 64 (4):1645-1655.
A Few Special Ordinal Ultrafilters.Claude Laflamme - 1996 - Journal of Symbolic Logic 61 (3):920-927.
Forcing and Stable Ordered–Union Ultrafilters.Todd Eisworth - 2002 - Journal of Symbolic Logic 67 (1):449-464.
Ultrapowers as Sheaves on a Category of Ultrafilters.Jonas Eliasson - 2004 - Archive for Mathematical Logic 43 (7):825-843.
On Milliken-Taylor Ultrafilters.Heike Mildenberger - 2011 - Notre Dame Journal of Formal Logic 52 (4):381-394.
Ultrafilters of Character $Omega_1$.Klaas Pieter Hart - 1989 - Journal of Symbolic Logic 54 (1):1-15.
Ultrafilters Which Extend Measures.Michael Benedikt - 1998 - Journal of Symbolic Logic 63 (2):638-662.
Ultrafilters on the Natural Numbers.Christopher Barney - 2003 - Journal of Symbolic Logic 68 (3):764-784.
A Note on Defining the Rudin-Keisler Ordering of Ultrafilters.Donald H. Pelletier - 1976 - Notre Dame Journal of Formal Logic 17 (2):284-286.

Analytics

Added to PP index
2013-11-23

Total views
23 ( #464,933 of 2,440,212 )

Recent downloads (6 months)
1 ( #432,124 of 2,440,212 )

How can I increase my downloads?

Downloads

My notes