Comments on bases in dependence structures

Abstract

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 B 1 and B 2 are bases of a dependence structure, then there is an injection σ: B 1 → B 2 such that ( B 2 / {σ( e )}) ∩ { e } is a basis for all e in B 1 . 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 B 1 and B 2 are bases of a dependence structure then given e in B 1 there is an f in B 2 such that both ( B 1 / { e }) ∩ { f } and ( B 2 / { f }) ∩ { e } are bases. This is a symmetrical kind of replacement theorem.