A Variation of Galvin and J\'onsson's approach to Sublattices of Free Lattices

TL;DR

This paper explores a new approach to understanding which lattices can be sublattices of free lattices, extending Galvin and Jonsson's work, with potential constructions of specific sublattices, aiming to solve an open problem in lattice theory.

Contribution

It proposes a novel strategy extending Galvin and Jonsson's characterization to construct sublattices with finite intervals and width, advancing the understanding of sublattices of free lattices.

Findings

Proposes a method to construct sublattices with finite proper subintervals.

Extends Galvin and Jonsson's characterization to new lattice classes.

Introduces explicit construction techniques for potential sublattices.

Abstract

This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by R.Freese, J.Jezek, and J.B. Nation published in 1995: "Which lattices (and in particular which countable lattices) are sublattices of a free lattice?" Despite partial progress over the decades, the problem is still unsolved. There is emphasis on the countable case because the current body of knowledge on sublattices of free lattices is most concentrated on when these sublattices are countably infinite. Galvin and Jonsson's paper characterized those sublattices of free lattices that are distributive by determining the distributive lattices which do not have any doubly reducible elements.. In this article, a possible strategy towards solving this open problem is proposed. I will do this by extending a portion of Galvin and Jonsson's paper, about 54 years after its publication. To the best of my knowledge, the results derived in this article are new and not mentioned elsewhere in the literature. Specifically, there appears to be a way to explicitly construct possible sublattices of free lattices in which: every proper subinterval is finite, the width of the lattice is finite. I will describe this approach I am developing in this article.