**MSC:**- 06A07 Combinatorics of partially ordered sets
- 05D05 Extremal set theory

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 fixed $k$ and $i$ our objective is to find a set $\mathcal{M}$ of

submatrices $M_1,M_2,\dots ,M_k$ such that $r(M_j)=i$ for all $j$ and the shadow of

$\mathcal{M}$ is minimal, that is, the number of submatrices $M$

with $r(M)=i-1$ contained in a member of $\mathcal{M}$ 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 finite sets: Let $A,B$ be

two disjoint subsets of a finite set $N$. We consider all subsets $F\subseteq N$

satisfying $A\not\subseteq F$ and $B\not\subseteq F$. For fixed $i,k$ we

determine a family $\mathcal{F}=\{F_1,\dots,F_k\}$ of such $i$--subsets of $N$

which has minimum number of $(i-1)$--sets contained in some $F\in\mathcal{F}$.

Finally, as an application we give a solution to the problem of finding an

ideal of given size and maximum weight in Submatrix Orders and in their

duals.