Flats and hyperplane arrangements for matroids with coefficients

TL;DR

This paper develops a comprehensive theory of flats and hyperplane arrangements for T-matroids over a tract, providing multiple equivalent descriptions and applications to tropical linear spaces.

Contribution

It introduces a unified framework for T-matroids, extending classical matroid concepts to the setting of tracts, and offers cryptomorphic characterizations including hyperplane and point-line arrangements.

Findings

Multiple cryptomorphic descriptions of T-matroids.

Application to tropical linear spaces and valuated matroids.

New insights into the structure of T-matroids over tracts.

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.