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, $\lambda \text{'}<\lambda$, and the ultrafilter $D$ is $\kappa$ -decomposable for all regular cardinals $\kappa$ with $\lambda \text{'}<\kappa <\lambda$. 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 $\mathrm{cf}\hspace{0.17em}\lambda$-decomposable or ${\lambda }^{+}$-decomposable. We also give applications to topological spaces and to abstract logics.