A module-theoretic approach to matroids

TL;DR

This paper develops a module-theoretic framework for matroids, connecting combinatorics, algebra, and tropical geometry, and introduces new concepts like a tropical reduced row echelon form.

Contribution

It systematically extends the idempotent module theory of matroids, providing new geometric, algebraic, and categorical insights and tools.

Findings

Geometric interpretation of strong matroid maps

A generalized notion of strong matroid maps via module homomorphisms

A tropical analogue of reduced row echelon form

Abstract

Speyer recognized that matroids encode the same data as a special class of tropical linear spaces and Shaw interpreted tropically certain basic matroid constructions; additionally, Frenk developed the perspective of tropical linear spaces as modules over an idempotent semifield. All together, this provides bridges between the combinatorics of matroids, the algebra of idempotent modules, and the geometry of tropical linear spaces. The goal of this paper is to strengthen and expand these bridges by systematically developing the idempotent module theory of matroids. Applications include a geometric interpretation of strong matroid maps and the factorization theorem; a generalized notion of strong matroid maps, via an embedding of the category of matroids into a category of module homomorphisms; a monotonicity property for the stable sum and stable intersection of tropical linear spaces; a novel perspective of fundamental transversal matroids; and a tropical analogue of reduced row echelon form.