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.