Extremality, Left-Modularity and Semidistributivity

TL;DR

This paper explores the relationships between three extended classes of lattices—join-semidistributive, join-extremal, and left-modular—establishing new implications and connecting these properties to topological concepts like lexicographic shellability.

Contribution

It proves that every join-semidistributive, left-modular lattice is join extremal, extending previous results and clarifying the interplay between these lattice properties.

Findings

Every join-semidistributive, left-modular lattice is join extremal.

Relations between lattice classes are clarified and extended.

Connections to lexicographic shellability are established.

Abstract

In this article we investigate the relations between three classes of lattices each extending the class of distributive lattices in a different way. In particular, we consider join-semidistributive, join-extremal and left-modular lattices, respectively. Our main motivation is a recent result by Thomas and Williams proving that every semidistributive, extremal lattice is left modular. We prove the converse of this on a slightly more general level. Our main result asserts that every join-semidistributive, left-modular lattice is join extremal. We also relate these properties to the topological notion of lexicographic shellability.