Polymatroids, closure operators and lattices

TL;DR

This paper explores the relationship between polymatroid closure operators and lattice theory, establishing a framework that generalizes matroid minors to lattice minors and connects graph minors to lattice minors.

Contribution

It introduces a lattice-theoretic perspective on polymatroid closure operators and defines a notion of minors for lattices enriched with generating sets, generalizing graph minors.

Findings

Polymatroid closure operators relate to lattices with generating sets similarly to matroids and geometric lattices.

Minors of the lattice of flats of a graphic matroid correspond to simple graph minors.

The lattice minor concept is generalized to all polymatroids.

Abstract

We study the closure operators of polymatroids from a lattice theoretic point of view. We show that polymatroid closure operators relate to lattices enriched with a generating set in the same way that matroids relate to geometric lattices. Through this relation we define a notion of minors for lattices enriched with a generating set. For the lattice of flats of a graphic matroid, the minors of the lattice are shown to correspond to simple minors of the graph when the vertices are labeled and the edges unlabeled. This correspondence is generalized to all polymatroids.