Kirchberger's Theorem for Complexes of Oriented Matroids

TL;DR

This paper generalizes Kirchberger's Theorem from rank 3 oriented matroids to complexes of oriented matroids, expanding its applicability within the theory of oriented matroids.

Contribution

It extends Kirchberger's Theorem to complexes of oriented matroids, a broader class than previously considered.

Findings

Proves Kirchberger's Theorem for complexes of oriented matroids

Generalizes previous results from rank 3 to broader complexes

Links separation theorems with foundational matroid theorems

Abstract

The separation theorem of Kirchberger can be proven using a combination of Farkas' Lemma and Caratheodory's Theorem. Since those theorems are at the heart of oriented matroids, we are interested in a generalization of Kirchberger's Theorem to them. This has already been done for rank 3 oriented matroids. Here we prove it for complexes of oriented matroids, which are a generalization of oriented matroids.