Generalizing Galvin and J\'onsson's Classification to N5

TL;DR

This paper extends Galvin and Jónsson's classification of distributive sublattices of free lattices to those within the variety generated by N5, revealing new structural properties and potential extensions to other semidistributive lattice varieties.

Contribution

It generalizes the classification to lattices in the N5 variety, identifying their structural properties and proposing extensions to other semidistributive lattice varieties.

Findings

Sublattices of free lattices in the N5 variety satisfy three key structural properties.

The results can be partially extended to lattices from seven known infinite semidistributive sequences.

Provides a framework for understanding sublattice classification beyond distributive cases.

Abstract

The problem of determining (up to lattice isomorphism) which lattices are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and J\'onsson when they classified (up to lattice isomorphism) all of the distributive sublattices of free lattices in 1959. In this paper, we weaken the requirement that of a free lattice be distributive to requiring that such a lattice belongs in the variety of lattices generated by the pentagon N5. Specifically, we use McKenzie's list of join-irreducible covers of the variety generated by N5 to extend Galvin and J\'onsson's results by proving that all sublattices of a free lattice that belong to the variety generated by N5 satisfy three structural properties. Afterwards, we explain how the results in this paper can be partially extended to lattices from seven known infinite sequences of semidistributive lattice varieties.