The free product of $q$-matroids

TL;DR

This paper introduces the concept of the free product of $q$-matroids, exploring its properties, maximality, irreducibility, uniqueness of factorization, and representability, especially for rank one uniform $q$-matroids.

Contribution

It defines the free product of $q$-matroids, analyzes its properties, and establishes uniqueness of factorization into irreducibles, extending matroid theory to the $q$-analogue.

Findings

The free product is maximal with respect to the weak order.

Irreducible $q$-matroids have a unique factorization.

Rank one uniform $q$-matroids are represented by clubs on the projective line.

Abstract

We introduce the notion of the free product of $q$-matroids, which is the $q$-analogue of the free product of matroids. We study the properties of this noncommutative binary operation, making an extensive use of the theory of cyclic flats. We show that the free product of two $q$-matroids $M_1$ and $M_2$ is maximal with respect to the weak order on $q$-matroids having $M_1$ as a restriction and $M_2$ as the complementary contraction. We characterise $q$-matroids that are irreducible with respect to the free product and we prove that the factorization of a $q$-matroid into a free product of irreducibles is unique up to isomorphism. We discuss the representability of the free product, with a particular focus on rank one uniform $q$-matroids and show that such a product is represented by clubs on the projective line.