The theory of saturated chain partitions of partial orders is applied to the minimum unichain covering problem in the product of partial orders (posets). Define the nested saturation property for a poset to be the existence of a sequence of chain partitions C 1 , C 2 ,... such that C k is k- and k + 1-saturated and the elements on chains of size at most k in C k contain the elements on chains of size at most (k - 1) in C k-1 . For the product of two posets P and Q with the nested saturation property, a unichain covering is constructed of size Σ Δ Pk Δ Qk , where d Pk is the size of the largest k-family in P and Δ k = d k - d k-1 . This is applied to prove that the largest semiantichain and smallest unichain covering have the same size for products of a special class of posets. © 1987.
West, D. B. (1987). Unichain coverings in partial orders with the nested saturation property. Discrete Mathematics, 63(2–3), 297–303. https://doi.org/10.1016/0012-365X(87)90018-5