A Generalization of a Theorem of Mandel

TL;DR

This paper generalizes Mandel's theorem, showing that the covector set of a conditional oriented matroid can be derived from its topes using face symmetry, facilitating geometric computations.

Contribution

It extends Mandel's theorem to conditional oriented matroids, introducing the face symmetry condition for covector determination from topes.

Findings

Covector sets can be derived using face symmetry in conditional oriented matroids.

The approach enables better computational representation of geometric configurations.

Application to apartments of hyperplane arrangements demonstrates practical utility.

Abstract

A theorem of Mandel allows to determine the covector set of an oriented matroid from its set of topes by using the composition condition. We provide a generalization of that result, stating that the covector set of a conditional oriented matroid can also be determined by its set of topes, but by using the face symmetry condition. It permits to represent geometrical configurations in terms of conditional oriented matroids, more suitable for computer calculations. We treat apartments of hyperplane arrangements as example.