Ineffability Properties of Pkλ

Abstract

<p>We first use some results of Menas to prove that every normal filter on PKλ extends the cub filter on PKλ thereby settling a basic question in the structure theory of filters on PKλ.</p> <p>Then we investigate ideal-theoretic and other aspects of ineffability properties of PKλ with particular emphasis on those which can be viewed as PKλ generalizations of weak compactness.</p> <p>In the course of these studies, we came to view mild λ-ineffability as a PKλ generalization of weak compactness in an ideal-theoretically weak sense, and sought a PKλ generalization of weak compactness in an ideal-theoretically stronger sense.</p> <p>To this end, we define the λ-Shelah property, a new ineffability property of PKλ between mild λ-ineffability and almost λ-ineffability, and prove results which support the contention that this is the property we sought.</p> <p>These results include characterizations of the λ-Shelah property in terms of a normal ideal on PKλ and if λ K.</p>