On the lattice of flats of a boolean representable simplicial complex

TL;DR

This paper characterizes the lattices of flats of boolean representable simplicial complexes, showing they are atomistic and semimodular iff the complex is a matroid, and introduces a construction for finite atomistic lattices.

Contribution

It provides a canonical construction for finite atomistic lattices and characterizes when these are lattices of flats of boolean representable simplicial complexes.

Findings

Lattices of flats are always atomistic.

Semimodularity occurs iff the complex is a matroid.

Every finite lattice can be realized as a lattice of flats.

Abstract

It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic lattices, providing a characterization of the lattices of flats of boolean representable simplicial complexes and a decidability condition. We remark that every finite lattice occurs as the lattice of flats of some simplicial complex.