In a graph, the class of all the sets of vertices that can be covered by some matching forms a matroid.
I wonder what kind of structure the class of all the matchings in a graph can have? Or does it not have any usual structure?
It seems to me that it is close to but not a matroid. Is it the reason to study the class of all the sets of vertices covered by some matching, instead of studying directly the class of all the matchings?