Weak Distributivity Implying Distributivity

Abstract

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