Frobenius and Sylvester inequalities for the chainable rank

This paper shows that any nonnegative $n \times m$ 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.