  The expansion axiom of matroids requires only the existence of some kind of
independent sets, not the uniqueness of them. This causes that the base
families of some matroids can be reduced while the unions of the base families
of these matroids remain unchanged. In this paper, we define unique expansion
matroids in which the expansion axiom has some extent uniqueness; we define
union minimal matroids in which the base families have some extent minimality.
Some properties of them and the relationship between them are studied. First,
we propose the concepts of secondary base and forming base family. Secondly, we
propose the concept of unique expansion matroid, and prove that a matroid is a
unique expansion matroid if and only if its forming base family is a partition.
Thirdly, we propose the concept of union minimal matroid, and prove that unique
expansion matroids are union minimal matroids. Finally, we extend the concept
of unique expansion matroid to unique exchange matroid and prove that both
unique expansion matroids and their dual matroids are unique exchange matroids.
