Pfaffian forms and Δ-matroids with coefficients

Abstract

In this paper I begin to build up a theory of Δ-matroids with coefficients parallel to the theory of matroids with coefficients. While the definition of a matroid with coefficients is based on the well-known Grassmann identity concerning determinants, a quite similar identity concerning Pfaffian forms leads to the theory of Δ-matroids with coefficients. Particular examples are Δ-matroids representable by some skew-symmetric matrix with coefficients in a field, oriented Δ-matroids, and valuated Δ-matroids.