ST-distributive and ST-modular lattices

Abstract

This thesis proposes two new classes of lattices: <em>ST</em>-distributive and <em>ST</em>-modular lattices. The idea is to define relative distributive and modular properties that are satisfied by some elements over two subsets <em>S</em> and <em>T</em> of a given lattice <em>L</em>. These new classes include the usual distributive and modular lattices. Our main results are (1) establishing some basic properties, (2) completely characterizing the maximal <em>S</em> and <em>T</em> (with some restrictions) to form <em>ST</em>-distributive lattices in the lattice families <strong>M</strong>_n and <strong>M</strong>_n,n, and (3) presenting an application of <em>ST</em>-modular lattices to convex sets. This application has been the first example of <em>ST</em>-modularity and the original motivation of our work [7]. All this extends our work in [10]. In addition, we discuss two other problems with new results. First, we present two shortcuts to the <strong>M</strong>₃-<strong>N</strong>₅ Theorem proof in [5, 4] and design two methods to compare the lengths of the three proofs [11]. Second, we introduce a poset on the cut-complexes of the 4-cube and show that it is a distributive lattice [9].