Matroids, hereditary collections and simplicial complexes having boolean representations

TL;DR

This paper develops a theory for representing hereditary collections, lattices, and simplicial complexes using boolean matrices, focusing on the lattice of flats and minimal elements.

Contribution

It introduces a novel representation framework connecting hereditary collections with boolean matrices and lattice theory, extending prior work by Izakhian and Rhodes.

Findings

Representation by boolean matrices corresponds to finite v-generated lattices.

The lattice of flats is central to the theory.

Minimal and strictly join irreducible elements are characterized.

Abstract

Inspired by the work of Izakhian and Rhodes, a theory of representation of hereditary collections by boolean matrices is developed. This corresponds to representation by finite $\vee$-generated lattices. The lattice of flats, defined for hereditary collections, lattices and matrices, plays a central role in the theory. The representations constitute a lattice and the minimal and strictly join irreducible elements are studied, as well as various closure operators.