A broad class of shellable lattices

TL;DR

This paper introduces modernistic and comodernistic lattices, proving their shellability and connecting these classes to known structures like solvable groups, thereby unifying various shellable lattice classes.

Contribution

It defines new lattice classes, proves their shellability, and links comodernistic lattices to solvable groups, unifying multiple shellable lattice families.

Findings

Modernistic and comodernistic lattices are shellable.

Subgroup lattices of solvable groups are comodernistic.

Solvability of a group is equivalent to its subgroup lattice being comodernistic.

Abstract

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k-equal partition lattices. We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To start with, the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable. We thus obtain a unified proof that lattice in these classes are shellable. Subgroup lattices of solvable groups form another family of comodernistic lattices that were already proved to be shellable. We show not only that subgroup lattices of solvable groups are comodernistic, but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice. Indeed, the definition of comodernistic exactly requires on every interval a lattice-theoretic analogue of the composition series in a solvable group. Thus, the relation between comodernistic lattices and solvable groups resembles, in several respects, that between supersolvable lattices and supersolvable groups.