The direct sum of $q$-matroids

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.