On the Product of Some Posets: Jump Number and Greediness

Abstract

Through combinatorial analysis we study the jump number, greediness and optimality of the products of chains, the product of an (upward rooted) tree and a chain. It is well known [1] that the dimension of products of $n$ chains is $n$. We construct a minimum realizer $L_1,\dots ,L_n$ for the products of $n$ chains such that $s(\bigcap _{i=1}^j{L}_i)\le s(\bigcap _{i=1}^{j+1}{L}_i)$ where $j=1,\dots ,n-1$.