A lattice framework for generalizing shellable complexes and matroids

TL;DR

This paper introduces a unified lattice framework for generalizing shellable complexes and matroids, extending classical concepts to power lattices and demonstrating new properties and constructions.

Contribution

It develops the notion of power lattices, unifies shellability across various complexes, and generalizes matroids within this new lattice framework.

Findings

Shellable P-complexes lead to shellable order complexes.

Construction of matroids on multiset subset lattices from weighted graphs.

Associated Stanley-Reisner rings are sequentially Cohen-Macaulay.

Abstract

We introduce the notion of power lattices that unifies and extends the equicardinal geometric lattices, Cartesian products of subspace lattices, and multiset subset lattices, among several others. The notions of shellability for simplicial complexes, q-complexes, and multicomplexes are then unified and extended to that of complexes in power lattices, which we name as P-complexes. A nontrivial class of shellable P-complexes are obtained via P-complexes of the independent sets of a matroid in power lattice, which we introduce to generalize matroids in Boolean lattices, q-matroids in subspace lattices, and sum-matroids in Cartesian products of subspace lattices. We also prove that shellable P-complexes in a power lattice yield shellable order complexes, extending the celebrated result of shellability of order complexes of (equicardinal) geometric lattices by Bj\"orner and also, a recent result on shellability of order complexes of lexicographically shellable q-complexes. Finally, we provide a construction of matroids on the lattice of multiset subsets from weighted graphs. We also consider a variation of Stanley-Reisner rings associated with shellable multicomplexes than the one considered by Herzog and Popescu and proved that these rings are sequentially Cohen-Macaulay.