A note on the normal filters extension property

TL;DR

This paper proves that under certain set-theoretic conditions, the extension property of normal filters to ultrafilters on a power set implies the same for all -complete filters on a set, answering a question posed by Gitik.

Contribution

It establishes a significant link between the extension properties of normal filters and all -complete filters, resolving a question in set theory.

Findings

Normal filter extension property implies -complete filter extension property under specified conditions

Answers a question of Gitik regarding filter extension properties

Provides insights into the structure of filters and ultrafilters in set theory

Abstract

We show that if $\lambda^{<\kappa} = \lambda$ and every normal filter on $P_\kappa\lambda$ can be extended to a $\kappa$-complete ultrafilter then so does every $\kappa$-complete filter on $\lambda$. This answers a question of Gitik.