Unichain coverings in partial orders with the nested saturation property (Q1086262)

In a direct product of posets unichains are chains that vary in only one coordinate. Semi-antichains are collections taking no pair of elements from a single unichain. A k-family in a poset P is the union of k antichains. The largest size of a k-family is denoted \(d_ k\). The number \(\Delta_ k=d_ k-d_{k-1}\) is also denoted \(\Delta^ P_ k\) for the poset P. A poset P has the nested saturation property if it has a sequence \(C=C_ 1,C_ 2,..\). of chain-partitions such that \(C_ k\) is \(k,k+1\) saturated and such that \(S_{k-1}(C_{k-1})\subseteq S_ k(C_ k)\), \(S_{\ell}(C_ j)\) is the set of elements of chains of size at most \(\ell\) in \(C_ j\). The main theorem of this paper is that: if P and Q have the nested saturation property then \(P\times Q\) has a unichain covering with \(\sum \Delta^ P_ k\Delta^ Q_ k\) chains, which, though not always minimal, is so in the case \(P_ n\times P_ m\) is the product of canonical poly-unsaturated posets, studied by the author as an important class of posets with respect to the Greene-Kleitman Theorem in his reference [4] that is reviewed in this journal [J. Comb. Theory, Ser. A 41, 105-116 (1986; Zbl 0583.06002)]. The dualization produces a largest cover of semi-antichains containing the same number of elements \(\sum \Delta^ P_ k\Delta^ Q_ k\) \((P=P_ n\), \(Q=P_ m)\) in this case as well.