K. Adaricheva | Searcharxiv

arXiv2025

Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond

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.

math.RA↗

arXiv2004

Join-semidistributive lattices of relatively convex sets

We give two sufficient conditions for the lattice Co(R^n,X) of relatively convex sets of n-dimensional real space R^n to be join-semidistributive, where X is a finite union of segments. We also prove that every finite lower bounded lattice can be embedded into Co(R^n,X), for a suitable finite subset X of R^n.

math.RA↗