On a theorem by Juh\'asz and Szentmikl\'ossy

TL;DR

This paper extends a theorem related to pseudocompactness to include singular cardinals and applies it to ultrafilter decomposability, broadening the understanding of ultrafilter products in set theory.

Contribution

It generalizes a theorem by Juhász and Szentmiklóssy to singular cardinals and explores implications for ultrafilter decomposability.

Findings

Extended theorem to singular cardinals.

Proved ultrafilter product decomposability for singular cardinals.

Connected pseudocompactness notions with ultrafilter properties.

Abstract

We extend a theorem by Juh\'asz and Szentmikl\'ossy to notions related to pseudocompactness. We also allow the case when one of the cardinals under consideration is singular. We give an application to the study of decomposable ultrafilters: if \kappa\ is singular, D is a uniform ultrafilter over \kappa ^+, and D' is a uniform ultrafilter over cf \kappa, then D' \times D is \kappa-decomposable.