Optimal shadows and ideals in submatrix orders

We consider the poset of all submatrices of a given matrix, ordered by containment. The unique rank function for this poset is given by r(M) = R(M)+C(M)?1, where R(M) and C(M) denote the number of rows and columns of a nonempty matrix M, respectively, and the rank of the empty matrix is 0. For xed k and i our objective is to nd a set M of submatrices M 1 ; M 2 ; : : :; M k such that r(M j) = i for all j and the shadow of M is minimal, that is, the number of submatrices M with r(M) = i ? 1 contained in a member of M should be smallest possible. Partial results concerning the shadow minimization problem for our poset were obtained by Sali. In general, he conjectured a theorem of Kruskal{Katona type to hold. We show that this conjecture is true. In fact, our result covers a slightly more general case (the cartesian product of a Submatrix Order and a Boolean lattice). The mentioned result can be formulated in terms of nite sets: Let A; B be two disjoint subsets of a nite set N. We consider all subsets F N satisfying A 6 6 F and B 6 6 F. For xed i; k we determine a family F = fF 1 ; : : :; F k g of such i{subsets of N which has minimum number of (i ? 1){sets contained in some F 2 F. Finally, as an application we give a solution to the problem of nding an ideal of given size and maximum weight in Submatrix Orders and in their duals. 1. Introduction In this paper we solve the Shadow Minimization Problem for the poset of submatrices of a matrix, i.e. we prove a theorem for this poset which is analogous to the classical Kruskal{Katona theorem 7, 8] for Boolean lattices. For all deenitions not included in this article we refer to Engel's book 2].