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

of a finite set $N$ which do not contain any of the $A_i$'s, where the

$A_i$'s are mutually disjoint subsets of $N$. The elements of $P$ are

ordered by inclusion. We show that $P$ belongs to the class of Macaulay

posets, i.e. we show a Kruskal-Katona type theorem for $P$.

For the case that the $A_i$'s form a partition of $N$, the dual $P^*$ of $P$

became known as the orthogonal product of simplices. Since the property of

being a Macaulay poset is preserved by turning to the dual, we show in

particular that orthogonal products of simplices are Macaulay posets.

Besides, we prove that the posets $P$ and $P^*$ are additive.