  Kinser developed a hierarchy of inequalities dealing with the dimensions of
certain spaces constructed from a given quantity of subspaces. These
inequalities can be applied to the rank function of a matroid, a geometric
object concerned with dependencies of subsets of a ground set. A matroid which
is representable by a matrix with entries from some finite field must satisfy
each of the Kinser inequalities. We provide results on the matroids which
satisfy each inequality and the structure of the hierarchy of such matroids.
