On the dimension of ordered sets with the 2-cutset property
A subset of an ordered set \(P\) is called a cutset for \(P\) if it intersects every maximal chain of \(P\). If for every element \(x\in P\) there exists a subset \(S\) of \(P\), \(| S|\leq n\) (\(n\) a nonnegative integer), the elements of which are noncomparable to \(x\) and \(\{x\}\cup S\) is a cutset for \(P\), then \(P\) is said to have the \(n\)-cutset property. The core of this article consists in investigations of the case \(n=2\) and the dimension of \(P\). It is shown that the dimension for ordered sets with the 2-cutset property is bounded by a nonnegative integer \(N\). An example of an ordered set with the 2-cutset property with dimension 4 is given. At present, it is the largest known dimension for these ordered sets. Another interesting result is the statement that each ordered set \(P\) with the 2-cutset property can be expressed as \(P= X\cup Y\), where \(X\) is an ordinal sum of intervals of \(P\) having dimension \(\leq 9\), and \(Y\) is a subset of \(P\) with the largest size of an antichain \(\leq 16\).
scientific article; zbMATH DE number 484784