A connection between decomposability of ultrafilters and possible cofinalities

Abstract

We introduce the decomposability spectrum $K_D=\{λ\geq ω| D \text{is} λ\text{-decomposable}\}$ of an ultrafilter $D$, and show that Shelah's $\pcf$ theory influences the possible values $K_D$ can take. For example, we show that if $\aaa$ is a set of regular cardinals, $μ\in \pcfa$, the ultrafilter $D$ is $|\aaa |^+$-complete and $K_D \subseteq \aaa$, then $μ\in K_D$. As a consequence, we show that if $ λ$ is singular and for some $ λ' < λ$ $K_D$ contains all regular cardinals in $ [λ', λ)$ then: (a) if $\cf λ= ω$ then either $ λ\in K_D$, or $ λ^+ \in K_D$; and (b) if $D$ is $(\cf λ)^+$-complete then $ λ^+ \in K_D$, and $\pp (λ)= λ^+$.