On a model for non commutative geometric spaces
type: Item
instance of: scholarly article
title: On a model for non commutative geometric spaces
zbMATH Open document ID: 0581.51008
DOI: 10.1007/BF01220477
author: Jochen Pfalzgraf
published in: Journal of Geometry
publication date: 1985-01-01T00:00:00Z
review text: The author proposes a model for non-commutative geometries, which allows to describe axioms by corresponding equations. The article constitutes the basis for further developments, although some first hand applications are made here, mainly to the theory previously developed by J. André et al. [cf. for example, \textit{J. André}, Geom. Dedicata 13, 351-360 (1983; Zbl 0515.05017)]. In the present article the author departs from a triple \((X,<,>,R),\) called non commutative geometric space, where X, R are non- empty sets and \(<,>: X^ 2\to R\) is a map. X is thought as the set of ''points'', the image of \(<,>\) being the set of directions (Richtungen) or ideal points (Fernpunkte) and \(<,>\) being called the ''parallel map''. Suitable axioms are introduced and their behaviour under morphisms is described. Further applications to group spaces are promised.
Mathematics Subject Classification ID: 51D15
zbMATH DE Number: 3929831
zbMATH Keywords: join operation
zbMATH Keywords: base point
zbMATH Keywords: connection line
zbMATH Keywords: Tamaschke condition
zbMATH Keywords: Veblen condition
zbMATH Keywords: skew-affine space
zbMATH Keywords: group space
zbMATH Keywords: model for non-commutative geometries
zbMATH Keywords: non commutative geometric space
zbMATH Keywords: directions
zbMATH Keywords: ideal points
zbMATH Keywords: Fernpunkte
reviewed by: Salvador D. Gigena
MaRDI profile type: Publication
cites work: Coherent configurations and noncommutative spaces
cites work: Q3859776
cites work: Q5659683
cites work: Q3681464
cites work: Q5671305
description: scientific article; zbMATH DE number 3929831