Polar graphs and corresponding involution sets, loops and Steiner triple systems.
type: Item
instance of: scholarly article
title: Polar graphs and corresponding involution sets, loops and Steiner triple systems.
zbMATH Open document ID: 1116.20043
author: Silvia Pianta
author: Elena Zizioli
author: Helmut Karzel
published in: Results in Mathematics
publication date: 2007-01-23T00:00:00Z
review text: This paper is part of the authors' investigations on regular involution sets \((P,\mathcal I)\), complete graphs with parallelism \((P,\mathcal E,\parallel)\) and involutorial difference loops \((P,+)\). After the section of notations and known results, the authors study characteristic configurations, graphs with polar triangles, substructures and automorphisms, and invariant polar involution sets and corresponding graphs and loops. In the last section they make the further assumption that the polar involution set \((P,\mathcal I)\) is invariant. In the paper there are the results: the polar graph \((P,\mathcal E,\parallel)\) satisfies the trapezium axiom, is self-distributive, \((P,+)\) is a Moufang loop, is an affine triple system; and: \((P,\mathcal E,\parallel)\) satisfies the quadrangle axiom, is a group, is an affine space.
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 51E10
Mathematics Subject Classification ID: 51A15
zbMATH DE Number: 5117891
zbMATH Keywords: Moufang loops
zbMATH Keywords: pseudo-affine spaces
zbMATH Keywords: planes
zbMATH Keywords: involutorial difference loops
zbMATH Keywords: graphs with parallelism
zbMATH Keywords: regular involution sets
reviewed by: C. Pereira da Silva
MaRDI profile type: Publication
full work available at URL: s00025-006-0214-4
OpenAlex ID: W2083081893
DOI: 10.1007/S00025-006-0214-4
description: scientific article; zbMATH DE number 5117891