The direct sum of $q$-matroids

TL;DR

This paper introduces a novel definition of the direct sum for $q$-matroids, extending a fundamental operation from classical matroid theory to the quantum analogue using submodular functions.

Contribution

It defines the direct sum for $q$-matroids using submodular functions and $q$-matroid union, establishing desirable properties for this new operation.

Findings

The direct sum for $q$-matroids is successfully defined.

The proposed definition retains key properties of classical direct sums.

The approach uses submodular functions and $q$-matroid union techniques.

Abstract

For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for $q$-matroids, the $q$-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the $q$-analogue of matroid union we come to a definition of the direct sum of $q$-matroids. As a motivation for this definition, we show it has some desirable properties.