The incidence structure of subspaces with well-scaled frames

Abstract In this paper, the incidence structure of classes of subspaces that generalize the regular (unimodular) subspaces of rational coordinate spaces is studied. Let F the a field and S ⊂- F β {0}. A subspace, V, of a coordinate space over F is S-regular if every elementary vector of V can be scaled by an element of F β {0} so that all of its non-zero entries are elements of S. A subspace that is {−1, +1 }-regular over the rational field is regular. Associated with a subspace, V, over an arbitrary (respectively, ordered) field is a matroid (oriented matroid) having as circuits (signed circuits) the set of supports (signed supports) of elementary vectors of V. Fundamental representation properties are established for the matroids that arise from certain classes of subspaces. Matroids that are (minor) minimally non-representable by various classes of subspaces are identified. A unique representability results is established for the oriented matroids of subspaces that are dyadic (i.e., {±20, ±21, ±22, …}-regular) over the rationals. A self-dual characterization is established for the matroids of S-regular subspaces which generalizes Minty's characterization of regular spaces as digraphoids.