Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

TL;DR

This paper investigates the structure of maximal sublattices in finite semidistributive lattices, focusing on the conjecture that their complements are always intervals, and provides complete descriptions for certain subclasses.

Contribution

It advances understanding of complements of maximal sublattices in semidistributive lattices, especially for convex geometries of dimension 2, and explores the conjecture about interval complements.

Findings

Complements of maximal sublattices in convex geometries of dimension 2 are fully characterized.

The conjecture holds for bounded lattices and is explored for broader classes.

Procedures for finding all complements of maximal sublattices are described.

Abstract

We study maximal sublattices of finite semidistributive lattices via their complements. We focus on the conjecture that such complements are always intervals, which is known to be true for bounded lattices. Since the class of semidistributive lattices is the intersection of classes of join- and meet-semidistributive lattices, we study also complements for these classes, and in particular convex geometries of convex dimension 2, which is a subclass of join-semidistributive lattices. In the latter case, we describe the complements of maximal sublattices completely, as well as the procedure of finding all complements of maximal sublattices.