Decomposable Ultrafilters and Possible Cofinalities

We use Shelah’s theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem 1. Suppose that λ is a singular cardinal, λ′ < λ, and the ultrafilter D is κ-decomposable for all regular cardinals κ with λ′ < κ < λ. Then D is either λ-decomposable, or λ-decomposable. Corollary 2. If λ is a singular cardinal, then an ultrafilter is (λ, λ)-regular if and only if it is either cf λ-decomposable or λdecomposable. We give applications to topological spaces and to abstract logics (Corollaries 8, 9 and Theorem 10). If F is a family of subsets of some set I, and λ is an infinite cardinal, a λ-decomposition for F is a function f : I → λ such that whenever X ⊆ λ and |X| < λ then {i ∈ I|f(i) ∈ X} 6∈ F . The family F is λ-decomposable if and only if there is a λ-decomposition for F . If D is an ultrafilter (that is, a maximal proper filter) let us define the decomposability spectrum KD of D by KD = {λ ≥ ω|D is λ-decomposable}. The question of the possible values the spectrum KD may take is particularly intriguing. Even the old problem from [P, Si] of characterizing those cardinals μ for which there is an ultrafilter D such that KD = {ω, μ} is not yet completely solved [Shr, p. 1007]. The case when KD is infinite is even more involved. [P] studied the situation in which λ is limit and KD ∩ λ is unbounded in λ; he found some assumptions which imply that λ ∈ KD. This is not always the case; if μ is strongly compact and cf λ < μ < λ then there is an ultrafilter D such that KD ∩ λ is unbounded in λ, and D is not λ-decomposable. If we are in the above situation, D is necessarily λ-decomposable (by [So, Lemma 3] and the proof of [P, Proposition