Symmetric incidence groupoids
type: Item
instance of: scholarly article
title: Symmetric incidence groupoids
zbMATH Open document ID: 1035.51010
author: Herbert Hotje
author: Silvia Pianta
author: Elena Zizioli
published in: Journal of Geometry
publication date: 2003-09-16T00:00:00Z
review text: Let \(X\) be the set of points and lines of the Euclidean (or hyperbolic or elliptic) plane. It is well known that each element \(x\) of \(X\) determines a unique reflection, say \(\tilde x\). Hence one can define a product by putting \(\tilde y\cdot \tilde x:=\tilde y\circ \tilde x\circ \tilde y\). Then \((\tilde X,\cdot)\) is an example of a reflection groupoid [cf. also \textit{H. Hotje}, \textit{M. Marchi} and \textit{S. Pianta}, Discrete Math. 129, No.1-3, 139-147 (1994; Zbl 0804.51018)]. In the present article, reflection groupoids are introduced in an axiomatic way. A reflection groupoid is said to be a symmetric incidence groupoid if its elements are at the same time the points of a linear space such that certain axioms of compatibility between the two structures are satisfied. The authors determine the connection between symmetric incidence groupoids and (commutative) kinematic spaces. Also the automorphisms of such structures are investigated.
Mathematics Subject Classification ID: 51J05
Mathematics Subject Classification ID: 20N20
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 1982544
zbMATH Keywords: symmetric incidence groupoid
zbMATH Keywords: reflection groupoids
zbMATH Keywords: kinematic spaces
zbMATH Keywords: automorphisms
reviewed by: Hans Havlicek
MaRDI profile type: Publication
full work available at URL: s00022-003-1141-0
OpenAlex ID: W2026259529
DOI: 10.1007/S00022-003-1141-0
description: scientific article; zbMATH DE number 1982544