On the Duality of Semiantichains and Unichain Coverings

Abstract

We study a min-max relation conjectured by Saks and West: For any two posets $P$ and $Q$ the size of a maximum semiantichain and the size of a minimum unichain covering in the product $P\times Q$ are equal. For positive we state conditions on $P$ and $Q$ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.