Frobenius and Sylvester Inequalities for the Chainable Rank

This paper shows that any nonnegative n × m matrix free of zero rows and columns determines a map of the partition lattice of the set of cardinality n into the partition lattice of the set of cardinality m. These maps have certain properties similar to those of linear maps on vector spaces. In particular, for such maps the rank function is correctly defined and possesses a number of properties of the ordinary rank, including an upper bound for the rank of a matrix product. However, so far no lower bound has been established. In this paper, the counterpart of the Frobenius inequality for the above rank function is proved and, as a corollary, the Sylvester bound, providing a lower bound for the rank of a matrix product, is obtained.