Axioms for maximal vectors of an oriented matroid; a combinatorial characterization of the regions determined by an arrangement of pseudohyperplanes

Abstract

We prove that an oriented matroid over a set E can be regarded as a subset W vertices of a cube [−1, +1] E′ > R E′ , E′ > E, symmetrical with respect to the origin of R E′ and with the property that every subset of vertices of W , lying on a face of the cube and symmetrical with respect to the center of the face, coincides with the orthogonal projection of W into the face. More precisely, if we are given a subset W of vertices of a cube [−1, +1] E′ > R E′ , E′ > E, satisfying the above symmetry property, then W is the family of maximal vectors (or maximal covectors) of an oriented matroid M over E, and E′ is the subset of elements of E which are not coloops (respectively, are not loops) of M. Actually, our main theorem gives the first set of axioms for maximal vectors of an oriented matroid which does not impose as an axiom hereditarity for minors. Using the Topological Representation Theorem for oriented matroids, this result may be regarded as a combinatorial characterization of the regions determined by an arrangement of pseudohyperplanes.