Fixed points of products and the strong fixed point property
type: Item
instance of: scholarly article
title: Fixed points of products and the strong fixed point property
zbMATH Open document ID: 0651.06001
DOI: 10.1007/BF00337886
author: Dwight Duffus
author: Norbert Sauer
published in: Order
publication date: 1987-01-01T00:00:00Z
review text: The paper is motivated by the well known open problem: If ordered sets X and Y both have the fixed point property (fpp), will their product XY also have the fixed point property? The authors introduce what they call the strong fixed point property: An ordered set X has the strong fixed point property if there is an order preserving map \(\Phi\) of \(X^ X\) to X such that \(\Phi\) (f) is a fixed point of f for all \(f\in X^ X\). Such a map \(\Phi\) is also called a selection map for X. Examples of four classes of ordered sets with the strong fixed point property are given, and questions are raised about the extent of the strong fixed point property within the collection of ordered sets with the fixed point property. Question: Does the fixed point property imply the strong fixed point property in general? In the case of finite ordered sets? What is the relation between the strong fixed point property and the relational fixed point property (every order preserving multifunction has a fixed point)? Is there an example of a finite ordered set with the fpp that is not dismantlable? Some results: Let X and Y be ordered sets with the fixed point property. If at least one of X and Y has the strong fixed point property, then XY has the fpp. If both X and Y have the strong fixed point property, then XY has the strong fixed point property.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4061300
zbMATH Keywords: fixed point property
zbMATH Keywords: product
zbMATH Keywords: order preserving map
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00337886
OpenAlex ID: W2061149801
description: scientific article; zbMATH DE number 4061300