Cores, cutsets and the fixed point property
This paper introduces the concept of a core in a partially ordered set P. Cores are then used to study the fixed point property for partial orders P and for products of partial orders \(P\times Q\). Several extensions of fixed point theorems in the recent literature are obtained by efficient use of these cores. Definition: An ordered pair (A,B) of chains in P is a core in P, if \(x\leq y\) for all \(x\in A\) and for all \(y\in B.\) Some results: 1. If P is chain-complete and contains a cutset S such that every nonempty subset of S has a join or a meet in P, then P has the fixed point property (FPP). (Theorem 3). 2. If P or Q is chain-complete, Q has a cutset S as in 1., and both P and Q have the FPP, then \(P\times Q\) has the FPP. (Theorem 5). 3. If T is an ordered set with the FPP and \(\{P_ t:\) \(t\in T\}\) is a disjoint family of ordered sets with the FPP, then its ordered sum \(\cup \{P_ t:\) \(t\in T\}\) has the FPP. (Theorem 7).
scientific article; zbMATH DE number 3993646