Multifunctions and the fixed point property for products of ordered sets
type: Item
instance of: scholarly article
title: Multifunctions and the fixed point property for products of ordered sets
zbMATH Open document ID: 0591.06007
DOI: 10.1007/BF00337924
author: Aleksander Rutkowski
published in: Order
publication date: 1985-01-01T00:00:00Z
review text: \textit{J. M. Walker} [Discrete Math. 48, 275-288 (1984; Zbl 0546.06003)] attached to every ordered set (1) (E,\(\leq)\) a quasi-order (2) \((P(E,\leq),\leq_ W)\) of the power set PE of E be defining \(A\leq_ WB\) iff \(A\subset ideal B \& B\subset filter A\), for any A,B\(\subset E\). The present author says that a multifunction F: (1)\(\to (M,\leq_ M)\) \((:=any\) ordered set) is order-preserving or increasing iff \(x\leq y\) implies \(Fx\leq_ WFy\) [cf. \textit{R. E. Smithson}, Proc. Am. Math. Soc. 28, 304-310 (1971; Zbl 0238.06003)] and (1) has GFPP \((:=general\) fixed point property) iff (2) has FPP. The main aim of the author was to improve Theorem 7.4 of Walker, loc. cit., by removing the property of chain completeness of factors and proving Theorem 1: If one of the ordered sets A,B has GFPP and the other one has FPP, then their product \(A\times B\) has FPP; Theorem 2: (1) is chain complete and for every increasing multifunction F: (1)\(\to (M,\leq_ M)\) there are comparable points \(e_ 0,e_ 1\in (1)\) such that \(e_ 1\in Fe_ 0\), then FPP of \((M,\leq_ M)\) implies the FPP of the product \(E\times M\).
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 3948342
zbMATH Keywords: ordered set
zbMATH Keywords: quasi-order
zbMATH Keywords: power set
zbMATH Keywords: multifunction
zbMATH Keywords: general fixed point property
zbMATH Keywords: chain completeness
zbMATH Keywords: product
Wikidata QID: Q93591404
MaRDI profile type: Publication
cites work: Fixed points in partially ordered sets
cites work: A characterization of complete lattices
cites work: On a fixed point theorem for partially ordered sets
cites work: The Problem of Fixed Points in Ordered Sets
cites work: Fixed Points of Order Preserving Multifunctions
cites work: Isotone relations and the fixed point property for posets
full work available at URL: https://doi.org/10.1007/bf00337924
OpenAlex ID: W2042095619
description: scientific article; zbMATH DE number 3948342