Comments on bases in dependence structures

Dependence structures (in the finite case, matroids) arise when one tries to abstract the properties of linear dependence of vectors in a vector space. With the help of a theorem due to P. Hall and M. Hall, Jr concerning systems of distinct representatives of families of finite sets, it is proved that if B1 and B2 are bases of a dependence structure, then there is an injection σ: B1 → B2 such that (B2 / {σ(e)}) ∩ {e} is a basis for all e in B1. A corollary is the theorem of R. Rado that all bases have the same cardinal number. In particular, it applies to bases of a vector space. Also proved is the fact that if B1 and B2 are bases of a dependence structure then given e in B1 there is an f in B2 such that both (B1 / {e}) ∩ {f} and (B2 / {f}) ∩ {e} are bases. This is a symmetrical kind of replacement theorem.