Fixed point property for 11-element sets
type: Item
instance of: scholarly article
title: Fixed point property for 11-element sets
zbMATH Open document ID: 0796.06002
DOI: 10.1007/BF01108828
published in: Order
publication date: 1994-09-28T00:00:00Z
review text: A poset \(P\) is said to have the fixed point property iff each order- preserving map \(f: P\to P\) has a fixed point. An element \(x\in P\) is irreducible in \(P\) iff it has a unique upper cover or a unique lower cover in \(P\). Let \(P\) be a poset and let \(a,b\in P\). Then \(a\) is called retractable (to \(b\)) iff \(a\neq b\) and \((\downarrow a)\backslash \{a\}\subseteq \downarrow b\) and \((\uparrow a)\backslash\{a\}\subseteq \uparrow b\). Main theorems: 1. Let \(P\) be a poset and let \(a\in P\) be retractable to \(b\in P\). Then \(P\) has the fixed point property iff (1) \(P\backslash\{a\}\) has the fixed point property; (2) One of \((\uparrow a)\backslash\{a\}\) and \((\downarrow a)\backslash\{a\}\) has the fixed point property; 2. All posets with 11 elements that have the fixed point property and no retractable point can be described in 8 figures. Finally, the author introduces some lemmas on 11-element posets with no retractable points.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 549272
zbMATH Keywords: irreducible point
zbMATH Keywords: fixed point property
zbMATH Keywords: retractable point
zbMATH Keywords: 11-element posets
author: Bernd Schröder
reviewed by: Bohumil František Šmarda
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01108828
OpenAlex ID: W133989269
description: scientific article; zbMATH DE number 549272