More on regular and decomposable ultrafilters in ZFC

✍ Scribed by Paolo Lipparini

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 363 KB
• Volume: 56
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200910013

✦ Synopsis

Abstract

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them:

(a) If m ≥ 1 and the ultrafilter D is (~m~(λ^+n^), ~m~(λ^+n^))‐regular, then D is κ ‐decomposable for some κ with λ ≤ κ ≤ 2^λ^ (Theorem 4.3(a^'^)).

(b) If λ is a strong limit cardinal and D is (~m~(λ^+n^), ~m~(λ^+n^))‐regular, then either D is (cf λ, cf λ)‐regular or there are arbitrarily large κ < λ for which D is κ ‐decomposable (Theorem 4.3(b)).

(c) Suppose that λ is singular, λ < κ, cf κ ≠ cf λ and D is (λ^+^, κ)‐regular. Then:

(i) D is either (cf λ, cf λ)‐regular, or (λ^'^, κ)‐regular for some λ^'^ < λ (Theorem 2.2).

(ii) If κ is regular, then D is either (λ, κ)‐regular, or (ω, κ^'^)‐regular for every κ^'^ < κ (Corollary 6.4).

(iii) If either (1) λ is a strong limit cardinal and λ^<λ^ < 2^κ^, or (2) λ^<λ^ < κ, then D is either λ ‐decomposable, or (λ^'^, κ)‐regular for some λ^'^ < λ (Theorem 6.5).

(d) If λ is singular, D is (μ, cf λ)‐regular and there are arbitrarily large ν < λ for which D is ν ‐decomposable, then D is κ ‐decomposable for some κ with λ ≤ κ ≤ λ^<μ^ (Theorem 5.1; actually, our result is stronger and involves a covering number).

(e) D × D^'^ is (λ, μ)‐regular if and only if there is a ν such that D is (ν, μ)‐regular and D^'^ is (λ, ν^'^)‐regular for all ν^∼^ < ν (Proposition 7.1).

We also list some problems, and furnish applications to topological spaces and to extended logics (Corollar‐ies 4.6 and 4.8) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)