Locally finite sublattices of free lattices

TL;DR

This paper proves that all locally finite sublattices of free lattices are countable, simplifying the complex problem of identifying which infinite lattices can be sublattices of free lattices.

Contribution

It establishes that all locally finite sublattices of free lattices are countable, reducing the broader problem of classifying sublattices.

Findings

All locally finite sublattices of free lattices are countable.

The proof uses a result from Baldwin, Berman, Glass, and Hodges on free algebras.

Simplifies the problem of characterizing sublattices of free lattices.

Abstract

The problem of determining which infinite lattices are (isomorphic to) sublattices of free lattices is in general unsolved and extremely difficult. In this note, we reduce the problem by proving that all locally finite sublattices of free lattices are countable by using a result from Baldwin, Berman, Glass and Hodges on free algebras.