Decomposable Ultrafilters and Possible Cofinalities

Lipparini, Paolo

Notre Dame J. Formal Logic, Tome 49 (2008) no. 1, p. 307-312 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2008-07-15
Classification:  $λ-decomposable, (μ,λ)-regular (ultra)-filter,  cofinality of a partial order,  (productive) [μ,λ]-compactness,  03C20,  03E04,  03C95,  54D20