COMs: Complexes of Oriented Matroids

TL;DR

This paper introduces conditional oriented matroids (COMs), a new combinatorial structure generalizing oriented matroids, characterized by three axioms, and explores their realizability and connections to hyperplane arrangements and Coxeter complexes.

Contribution

It defines COMs as a new class of structures extending oriented matroids, with a binary composition scheme and realizability conditions, broadening the scope of combinatorial geometry.

Findings

COMs characterized by three cocircuit axioms

Every COM can be constructed from oriented matroids via a binary composition scheme

Realizable COMs correspond to hyperplane arrangements in convex sets

Abstract

In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of "conditional oriented matroid" (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces. A realizable COM is represented by a hyperplane arrangement restricted to an open convex set. Among these are the examples formed by linear extensions of ordered sets, generalizing the oriented matroids corresponding to the permutohedra. Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non-positively curved Coxeter zonotopal complexes give rise to locally realizable COMs.