The defect in an invariant reflection structure
type: Item
instance of: scholarly article
title: The defect in an invariant reflection structure
zbMATH Open document ID: 1232.51012
author: Helmut Karzel
author: Mario Marchi
author: Silvia Pianta
published in: Journal of Geometry
publication date: 2011-10-25T00:00:00Z
review text: The pair \((P,I)\) is called an \textit{involution set} if \(P\neq\emptyset\) is a set and \(I\subseteq J := \{\sigma\in SymP \mid \sigma^2 = id \}\). For \(a, b \in P\) and \(\alpha,\beta \in \langle I\rangle\), let \(\widetilde{a,b} := \{\iota \in I \mid \iota (a)=b \}\) and \(\tilde{a} :=\widetilde {a, a}\). An involution set \((P,I)\) is called a \textit{reflection structure} if it is regular, i.e. if \(\forall a,b \in P:| \widetilde{a,b} | = 1\), and \textit{invariant} if \(\forall\alpha \in I : \alpha\circ I \circ \alpha = I\). For an invariant reflection structure \((P,I)\) the authors define the following two \textit{defect functions}, \[ \rho: P^3 \rightarrow SymP; (a,b,c) \mapsto \rho_{a;b,c} := \widetilde{a,c} \circ \widetilde{b,c} \circ \widetilde{a,b} \] and \[ \delta : P^3 \rightarrow SymP; (a,b,c) \mapsto \delta_{a;b,c} := \tilde{a} \circ \rho_{a;b,c}. \] If \(\delta\) is constant, then the reflection structure \((P,I)\) is called \textit{singular}; otherwise it will be called \textit{ordinary}. The authors show: if the invariant reflection structure \((P,I)\) is singular, then the set \(I \circ I\) := \(\{\alpha \circ \beta \mid \alpha, \beta \in I\}\) is a commutative regular subgroup of \(SymP\). They investigate invariant ordinary reflection structures \((P,I)\) in detail. First they introduce several three reflection axioms, then they define various blocks and determine relations between them. Let \(o \in P\), put \(P^\ast:=P\setminus\{o\}\) and let \((P,+)\) be the K-loop derived from \((P,I)\) in \(o\). The authors introduce two centralizers on \((P^\ast)^2\). They study the representation of blocks by centralizers and exchange conditions for centralizers. Finally they give necessary and sufficient conditions for some incidence structures to be linear spaces.
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 5963285
zbMATH Keywords: reflection structure
zbMATH Keywords: loop derivation
zbMATH Keywords: loop
zbMATH Keywords: \(K\)-loop
zbMATH Keywords: defect
MaRDI profile type: Publication
full work available at URL: s00022-010-0058-7
OpenAlex ID: W2030776395
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DOI: 10.1007/S00022-010-0058-7
description: scientific article; zbMATH DE number 5963285