Ordonnés escamotables et points fixes. (Dismantlable orders and fixed points)
type: Item
instance of: scholarly article
title: Ordonnés escamotables et points fixes. (Dismantlable orders and fixed points)
zbMATH Open document ID: 0591.06008
DOI: 10.1016/0012-365X(85)90126-8
author: Julien Constantin
author: Gilles Fournier
published in: Discrete Mathematics
publication date: 1985-01-01T00:00:00Z
review text: For any ordered set (E,\(\leq)\) the authors consider the set of all selfmappings on E, each preserving the comparability relation \(\sim\) on (E,\(\leq)\), i.e. \(\sim:=\leq \cup \geq\). Inspired by the approach and results of \textit{K. Baclawski} and \textit{A. Björner} [Adv. Math. 31, 263- 287 (1979; Zbl 0417.06002)], they prove theorem 1.1: If E is finite and \(f: E\to E\) preserves \(\sim\), then \(\Lambda (t)=\chi (M_ f)\) (for f increasing [decreasing] this was proved by Baclawski-Björner, loc. cit. Theorems 1.1, 1.2). Theorem 5.2: Every left [right] badly complemented finite (E,\(\leq)\) is contractible and has the f.p.p. (fixed point property); this is a nice generalisation of a nice \textit{K. Baclawski} and \textit{A. Björner} result [J. Comb. Theory, Ser. A 30, 335-338 (1981; Zbl 0459.06004)] that for every finite non complemented lattice L its part \(L\setminus \{0,1\}\) has the f.p.p. Definition: (E,\(\leq)\) is said to be left [right] badly complemented if, for some \(x\in E\) and every \(y\in E\setminus \{x\}\) the set xly [xry] is contractible. For x,y\(\in E\), \(x\neq y\) one puts xly\([xry]:=\{v_{n-1}|\) \(x=v_ 0\geq [\leq]v_ 1\sim v_ 2\sim...\sim v_{n-1}\sim v_ n=y\), n minimal\(\}\).
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 3948343
zbMATH Keywords: ordered set
zbMATH Keywords: selfmappings
zbMATH Keywords: comparability relation
zbMATH Keywords: contractible
zbMATH Keywords: fixed point property
zbMATH Keywords: finite non complemented lattice
zbMATH Keywords: badly complemented
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3948343