A connection between decomposable ultrafilters and possible cofinalities. II

Abstract

We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $\lambda$ is a singular cardinal, ${\lambda }^{\prime }<\lambda$, and the ultrafilter $D$ is $\kappa$-decomposable for all regular cardinals $\kappa$ with ${\lambda }^{\prime }<\kappa <\lambda$. Then $D$ is either $\lambda$-decomposable, or ${\lambda }^{+}$-decomposable. We give applications to topological spaces and to abstract logics.

6 pages