More on regular and decomposable ultrafilters in ZFC

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)