Extremality in semidistributive lattices

TL;DR

This paper explores the properties and interrelations of extremal, left modular, congruence uniform, and semidistributive lattices, providing new characterizations, counterexamples, and dimension formulas that extend existing lattice theory results.

Contribution

It offers new characterizations of left modular lattices, establishes conditions for extremality, and generalizes the dimension formula for semidistributive extremal lattices.

Findings

A new characterization of left modular lattices via edge-labellings.

A proof that a congruence uniform lattice is shellable iff it is extremal.

A counterexample showing subcomplexes of canonical join complexes need not be such complexes.

Abstract

We establish several independent results concerning extremal, left modular, congruence uniform, and semidistributive lattices. An equivalent characterization of left modular lattices is obtained in terms of edge-labellings, together with necessary and sufficient conditions on the doubling steps in the construction of congruence normal lattices that ensure left modularity or extremality. We prove that a congruence uniform lattice is shellable if and only if it is extremal. We answer a question of Barnard by constructing a counterexample showing that an induced subcomplex of a canonical join complex need not itself be such a complex. Finally, we show that the order dimension of a semidistributive extremal lattice equals the chromatic number of the complement of its Galois graph, generalizing a theorem of Dilworth for distributive lattices. As an application, we determine the dimensions of generalizations of the Hochschild lattice, of the parabolic Tamari lattice, and of some lattices of torsion classes.