Potential Methods for Extending Galvin and J\'onsson's Characterization of Distributive Sublattices of Free Lattices

TL;DR

This paper revisits Galvin and Jf3nsson's 1959 characterization of distributive sublattices of free lattices, offering new proofs, exploring potential generalizations, and addressing a long-standing open problem in lattice theory.

Contribution

It provides new proofs of existing results, introduces the concept of spanning pairs for potential generalizations, and offers insights into finitely generated lattices over semidistributive varieties.

Findings

New proofs of Galvin and Jf3nsson's results

Partial results on spanning pairs and finite width sublattices

Observations on finitely generated lattices over semidistributive varieties

Abstract

In 1959, F.Galvin and B.Jonsson characterized distributive sublattices of free lattices in their paper. In this paper, I will create new proofs to a portion of Galvin and J\'onsson's results. Based on these new proofs, I will explore possible generalizations of F.Galvin and B.J\'onsson's work by defining \emph{spanning pairs} and proving partial results which may help with analysing finite width sublattices of free lattices; and by making some new observations on finitely generated lattices over semidistributive varieties. The work done in this paper may assist in attacking the following long-standing open problem: Which countable lattices are isomorphic to a sublattice of a free lattice?