Affine geometry over free unitary modules
type: Item
instance of: scholarly article
title: Affine geometry over free unitary modules
zbMATH Open document ID: 0581.51007
DOI: 10.1007/BF01220474
author: Rainer Severin
author: Klaus Wolf
author: Werner Leissner
published in: Journal of Geometry
publication date: 1985-01-01T00:00:00Z
review text: Let P be a set and L be a non-empty set of non-empty subsets of P. The elements of P are called ''points'' and those of L ''lines''. It is also assumed the existence of an antireflexive relation, non-neighboured, on P, and of an equivalence relation and parallelism on L, denoted by \(\phi\), \(\|\), respectively. The authors introduce appropriate geometrical axioms on the quadruplet \((P,L,\phi,\|)\) in order to define successively the concepts of generalized affine incidence structure (GAIS), generalized affine spaces (GAS), and generalized affine translation spaces (GATS), with a structure theorem being proven for this last. The main theorem reads as follows: The class of all affine Barbilian-spaces coincides with the class of all GATS which satisfy, in addition, the stretching-axiom, namely: for all p, q, r in P, with \(p\neq q\) and r in \(\overline{pq}\), there exists a p-dilatation mapping q onto r.
Mathematics Subject Classification ID: 51C05
zbMATH DE Number: 3929830
zbMATH Keywords: Barbilian domain
zbMATH Keywords: translation structure
zbMATH Keywords: crossline basis
zbMATH Keywords: little axiom of Desargues
zbMATH Keywords: non-neighboured
zbMATH Keywords: parallelism
zbMATH Keywords: incidence structure
zbMATH Keywords: affine Barbilian-spaces
reviewed by: Salvador D. Gigena
MaRDI profile type: Publication
cites work: Ebene Geometrie über einem Ring
cites work: On Barbilian domains over commutative rings
cites work: Q5639812
cites work: Q5662517
cites work: Affine Barbilian-Ebenen. I
cites work: Parallelodromie-Ebenen
cites work: Affine Barbilian structures
full work available at URL: https://doi.org/10.1007/bf01220474
OpenAlex ID: W2025967252
description: scientific article; zbMATH DE number 3929830