Extremality, Left-Modularity and Semidistributivity
arXiv:2112.07959 · doi:10.1007/s00012-023-00814-8
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.
10 pages, 5 figures. Comments are welcome