Games on base matrices

TL;DR

This paper explores the properties of base matrices in the context of distributivity, revealing that certain base matrices must have non-cofinal maximal branches when their height exceeds a specific regular cardinal.

Contribution

It introduces a game-theoretic characterization of distributivity and applies it to analyze base matrices for the quotient algebra ( extomega)/fin, establishing new structural constraints.

Findings

Base matrices of regular height > ( extomega) have non-cofinal maximal branches.

Game characterization effectively analyzes distributivity properties.

Structural limitations of base matrices are identified.

Abstract

Using a game characterization of distributivity, we show that base matrices for $\mathcal{P}(\omega)/\text{fin}$ of regular height larger than $\mathfrak{h}$ necessarily have maximal branches which are not cofinal.