Partial Steiner triple systems as bases of flexible algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1062.51003
author: Domenico Lenzi
publication date: 2005-06-22T00:00:00Z
Mathematics Subject Classification ID: 51E10
Mathematics Subject Classification ID: 17A20
Mathematics Subject Classification ID: 51C05
zbMATH DE Number: 1604438
zbMATH Keywords: Cayley algebra
zbMATH Keywords: flexible algebra
zbMATH Keywords: Steiner system
MaRDI profile type: Publication
Recommended article: Q5203196
Recommended article: Q4276196
Recommended article: Die Varietät der auflösbaren Ternare der Ordnung 2. (The variety of soluble Steiner ternary systems of order 2)
Recommended article: Quadratic algebras from steiner triple systems
Recommended article: Q5471949
Recommended article: C-LOOPS: EXTENSIONS AND CONSTRUCTIONS
Recommended article: Q5021086
Recommended article: Q4721143
Recommended article: Sur les surfaces algèbriques de genres arithmétique et géométrique zéro dont le genre linéaire est égal à deaux.
Recommended article: Q3743515
title: Partial Steiner triple systems as bases of flexible algebras
published in: PU.M.A. Pure Mathematics and Applications
review text: An algebra \(V\) is flexible if \((xy)x=x(yx)\), for all \(x,y\in V\). A basis \(B\) of \(V\) is called a Steiner basis if \(B\cup(-B)\cup\{0\}\) is closed under the algebra multiplication of \(V\), if either \(xy=yx\) or \(xy=-yx\) (excluding characteristic 2 here), and if for every \(x,y,\in B\), whenever \(xy\neq yx\) and \(xy\in B\), then \((xy)y\neq y(xy)\in\{x,-x\}\) and \(x(xy)\neq (xy)x\in \{y,-y\}\). Such a basis defines a partial Steiner triple system on \(B\) in the obvious way (2 elements are contained in at most one triple \(\{x,y,z\}\) with \(z=xy=-yx\)). The author proves a lot of small properties of flexible algebras (with and without a Steiner basis), and constructs such an algebra from any given abstract partial Steiner triple system. NEWLINENEWLINEThe motivation for such work is a bit unclear to me, but it certainly generalizes the representation of the 7 standard basis elements (except 1) of the classical Cayley division algebra over the real numbers using the lines of the projective plane of order 2 (the so-called Fano configuration) as noticed by \textit{H. Freudenthal} [Geom. Dedicata 19, 7--63 (1985; Zbl 0573.51004)].
reviewed by: Hendrik Van Maldeghem
description: scientific article; zbMATH DE number 1604438