More on regular and decomposable ultrafilters in ZFC
Article first published online: 12 JUL 2010
Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Mathematical Logic Quarterly
Volume 56, Issue 4, pages 340–374, August 2010
How to Cite
Lipparini, P. (2010), More on regular and decomposable ultrafilters in ZFC. Mathematical Logic Quarterly, 56: 340–374. doi: 10.1002/malq.200910013
- Issue published online: 12 JUL 2010
- Article first published online: 12 JUL 2010
- Manuscript Accepted: 30 SEP 2009
- Manuscript Revised: 17 SEP 2009
- Manuscript Received: 31 MAY 2009
- (λ, μ)-regular;
- κ -decomposable;
- λ -descendingly incomplete ultrafilter;
- sum of ultrafilters;
- cardinalities of ultrapowers.
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 ≤ 2λ (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 < 2κ, 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)