A Generalization of a Theorem of Mandel

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.