Convexity in ordered matroids and the generalized external order

TL;DR

This paper introduces a new convex geometry associated with ordered, unoriented matroids, extending the external order on matroid bases, and develops a generalized Tutte polynomial, enriching matroid theory with novel lattice structures.

Contribution

It constructs a convex geometry for ordered matroids using generalized activity, leading to a new representability concept and a refined external order lattice.

Findings

The generalized external order forms a supersolvable meet-distributive lattice.

The lattice refines the geometric lattice of flats.

A new trivariate generating function generalizing the Tutte polynomial is introduced.

Abstract

In 1980, Las Vergnas defined a notion of discrete convexity for oriented matroids, which Edelman subsequently related to the theory of anti-exchange closure functions and convex geometries. In this paper, we use generalized matroid activity to construct a convex geometry associated with an ordered, unoriented matroid. The construction in particular yields a new type of representability for an ordered matroid defined by the affine representability of its corresponding convex geometry. The lattice of convex sets of this convex geometry induces an ordering on the matroid independent sets which extends the external active order on matroid bases. We show that this generalized external order forms a supersolvable meet-distributive lattice refining the geometric lattice of flats, and we uniquely characterize the lattices isomorphic to the external order of a matroid. Finally, we introduce a new trivariate generating function generalizing the matroid Tutte polynomial.