Defining the q-analogue of a matroid

This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) crypto-morphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid. This paper establishes the definition and several basic properties of q-matroids. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid. We give definitions of a q-matroid in terms of its rank function and independent spaces. The dual, restriction and contraction of a q-matroid are defined, as well as truncation, closure, and circuits. Several definitions and results are straightforward translations of facts for ordinary matroids, but some notions are more subtle. We illustrate the theory by some running examples and conclude with a discussion on further research directions involving q-matroids. Many theorems in this article have a proof that is a straightforward q-analogue of the proof for the case of ordinary matroids. Although this makes them appear very easy, we feel it is needed to include them for completeness and also because it is not a guarantee that q-analogues of proofs exist.