Matchings in matroids over abelian groups

We formulate and prove matroid analogues of results concerning matchings in groups. A matching in an abelian group \((G,+)\) is a bijection \(f:A\rightarrow B\) between two finite subsets A, B of G satisfying \(a+f(a)\notin A\) for all \(a\in A\). A group G has the matching property if for every two finite subsets \(A,B \subset G\) of the same size with \(0 \notin B\), there exists a matching from A to B. In Losonczy (Adv Appl Math 20(3):385–391, 1998) it was proved that an abelian group has the matching property if and only if it is torsion-free or cyclic of prime order. Here we consider a similar question in a matroid setting. We introduce an analogous notion of matching between matroids whose ground sets are subsets of an abelian group G, and we obtain criteria for the existence of such matchings. Our tools are classical theorems in matroid theory, group theory and additive number theory.