More Results on Regular Ultrafilters in Zfc

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 (im(λ),im(λ))regular then D is κ-decomposable for some κ with λ ≤ κ ≤ 2 (Theorem 4.3(a)). (b) If λ is a strong limit cardinal and D is (im(λ ),im(λ ))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 6= cf λ and D is (λ, κ)-regular. Then: (i) D is either (cf λ, cfλ)-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 < 2, or (2) 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, cfλ)-regular and there are arbitrarily large ν < λ for which D is ν-decomposable then D is κdecomposable for some κ with λ ≤ κ ≤ λ (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