On the Product of Some Posets: Jump Number and Greediness

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 \(L_1, \ldots, L_n\) for the products of \(n\) chains such that \(s(\bigcap_{i=1}^{j}L_i) \leq s(\bigcap_{i=1}^{j+1}L_i)\) where \(j = 1, \ldots, n-1\).