A new proof of Funayama's theorem

TL;DR

This paper offers a new, purely order-theoretic proof of Funayama's theorem, which characterizes when a lattice can be embedded into a complete Boolean algebra while preserving joins and meets.

Contribution

It introduces a novel proof approach and extends Funayama's theorem to broader classes of lattices, enhancing understanding of lattice embeddings.

Findings

New order-theoretic proof of Funayama's theorem

Generalizations of the theorem for broader lattice classes

Clarification of conditions for lattice embeddings into Boolean algebras

Abstract

Funayama proved that a lattice embeds into a complete Boolean algebra in such a way that all existing joins and meets are preserved if and only if the lattice satisfies the join-infinite and meet-infinite distributive laws. There are several proofs of this classic result in the literature. In this note, we provide a new and purely order-theoretic proof of Funayama's theorem, as well as of generalizations of the theorem.