Flats and hyperplane arrangements for matroids with coefficients

Abstract

Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its lattice of T-flats, as a hyperplane arrangement over T, as point-line arrangements in projective space over T and as a quiver representation over T. We examplify these notions in the case of tropical linear spaces, a.k.a. valuated matroids.