Removing critical pairs
Let P be a poset. A pair (x,y) of incomparable elements of P is called critical if \(a<x\) implies \(a<y\), and \(b>y\) impliex \(b>x\) for all \(a,b\in P\). A pair (x,y) is said to be removable if its removal decreases the dimension by at most one. In the paper under review the author gives counterexamples to the following conjectures for posets with at least three elements. 1) If \(x\in P\) is contained in some critical pair, then there exists \(y\in P\setminus \{x\}\) such that (x,y) or (y,x) is a critical and removable pair. 2) Every critical pair is removable. The author believes that the well-known weaker conjecture is true: Every poset contains a removable pair. Here he describes how Ferrers dimension can be used in order to construct the counterexamples. In particular, removal theorems via Ferrers dimensions are proved.
scientific article; zbMATH DE number 4134105