Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

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.