Uwe Leck
On the orthogonal product of simplices and direct products of truncated Boolean lattices
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

06A07 Combinatorics of partially ordered sets

06A12 Semilattices, See also {20M10}

Abstract: The initial point of this paper are two Kruskal-Katona type theorems.
The first is the Colored Kruskal-Katona Theorem which can be stated as
follows: Direct products of the form $B_{k_1}^1\times B_{k_2}^1\times
\cdots\times B_{k_n}^1$ belong to the class of Macaulay posets, where
$B_k^t$ denotes the poset consisting of the $t+1$ lowest levels of the
Boolean lattice $B_k$. The second one is a recent result saying that
also the products $B_{k_1}^{k_1-1}\times B_{k_2}^{k_2-1}\times
\cdots\times B_{k_n}^{k_n-1}$ are Macaulay posets. The main result of
this paper is that the natural common generalization to products of
truncated Boolean lattices does not hold, i.e. that $(B_k^t)^n$ is a
Macaulay poset only if $t\in\{0,1,k-1,k\}$.
Keywords: Macaulay poset, Kruskal-Katona theorem, partially ordered set