Matroids

A finite set E with the properties:

(i) The empty set is independent

(ii) Every subset of an independent set is independent

(iii) If A and B are independent sets, with the number of elements of A being (at least) one less than the number of elements of B, then there is some x in B that is not in A such that A union {x} is also independent.

Property (iii) is the exchange axiom.

This idea may be applied to many vector as well as non-vector space problems.