Weak Distributivity Implying Distributivity

TL;DR

This paper demonstrates that certain weak distributivity conditions on complete Boolean algebras imply stronger distributivity properties, with implications for large cardinal assumptions.

Contribution

It establishes new conditions under which weak distributivity implies full distributivity in Boolean algebras, extending previous results.

Findings

Weak $(eth_eta, eta)$-distributivity implies $(eth_eta, 2)$-distributivity.

Weak $(2^ heta, heta)$-distributivity with large cardinal assumptions implies full $( heta, 2)$-distributivity.

Results connect weak distributivity conditions with large cardinal properties.

Abstract

Let $\mathbb{B}$ be a complete Boolean algebra. We show, as an application of a previous result of the author, that if $\lambda$ is an infinite cardinal and $\mathbb{B}$ is weakly $(\lambda^\omega, \omega)$-distributive, then $\mathbb{B}$ is $(\lambda, 2)$-distributive. Using a parallel result, we show that if $\kappa$ is a weakly compact cardinal such that $\mathbb{B}$ is weakly $(2^\kappa, \kappa)$-distributive and $\mathbb{B}$ is $(\alpha, 2)$-distributive for each $\alpha < \kappa$, then $\mathbb{B}$ is $(\kappa, 2)$-distributive.