A connection between decomposability of ultrafilters and possible cofinalities

Abstract

We introduce the decomposability spectrum $K_D=\{\lambda \ge \omega |D\text{is}\lambda \text{-decomposable}\}$ 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 $\mu \in K_D$. As a consequence, we show that if $\lambda$ is singular and for some ${\lambda }^{\prime }<\lambda$ $K_D$ contains all regular cardinals in $[{\lambda }^{\prime },\lambda )$ then: (a) if $\cf λ= ω$ then either $\lambda \in K_D$, or ${\lambda }^{+}\in K_D$; and (b) if $D$ is $(\cf λ)^+$-complete then ${\lambda }^{+}\in K_D$, and $\pp (λ)= λ^+$.

8 pages