Decomposable Ultrafilters and Possible Cofinalities

Abstract

We use Shelah's theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem: Suppose that $\lambda$ is a singular cardinal, $\lambda ' \lessthan \lambda$, and the ultrafilter $D$ is $\kappa$ -decomposable for all regular cardinals $\kappa$ with $\lambda '\lessthan \kappa \lessthan \lambda$. Then $D$ is either $\lambda$-decomposable or $\lambda ^+$-decomposable. Corollary: If $\lambda$ is a singular cardinal, then an ultrafilter is ($\lambda$,$\lambda$)-regular if and only if it is either $\operator{cf} \lambda$-decomposable or $\lambda^+$-decomposable. We also give applications to topological spaces and to abstract logics.