Cores, cutsets and the fixed point property
type: Item
instance of: scholarly article
title: Cores, cutsets and the fixed point property
zbMATH Open document ID: 0614.06002
DOI: 10.1007/BF00400289
author: Aleksander Rutkowski
published in: Order
publication date: 1986-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 3993646
zbMATH Keywords: partially ordered set
zbMATH Keywords: fixed point property
zbMATH Keywords: products of partial orders
zbMATH Keywords: cores
zbMATH Keywords: chain-complete
zbMATH Keywords: cutset
zbMATH Keywords: ordered sum
MaRDI profile type: Publication
cites work: A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems
cites work: Fixed points in partially ordered sets
cites work: On a fixed point theorem for partially ordered sets
cites work: The Problem of Fixed Points in Ordered Sets
cites work: Multifunctions and the fixed point property for products of ordered sets
cites work: Finite cutsets and finite antichains
description: scientific article; zbMATH DE number 3993646