A connection between decomposability of ultrafilters and possible cofinalities

Paolo Lipparini

Published 2006-04-08Version 1

We introduce the decomposability spectrum $K_D=\{\lambda \geq \omega| D \text{is} \lambda\text{-decomposable}\}$ 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, $\mu \in \pcfa$, 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' < \lambda $ $K_D$ contains all regular cardinals in $ [\lambda', \lambda)$ then: (a) if $\cf \lambda = \omega $ then either $ \lambda \in K_D$, or $ \lambda ^+ \in K_D$; and (b) if $D$ is $(\cf \lambda)^+$-complete then $ \lambda ^+ \in K_D$, and $\pp (\lambda)= \lambda ^+$.