Finite Cayley-Klein geometries.
type: Item
instance of: scholarly article
title: Finite Cayley-Klein geometries.
zbMATH Open document ID: 0608.51009
DOI: 10.1007/BF01189289
author: Horst Struve
author: Rolf Struve
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Finite geometry can be characterized as an interplay of combinatorical, geometric and algebraic ideas. There is no common-shared concept of finite metric geometry. In this article the definition of the real Cayley-Klein geometries [the authors, Z. Math. Logik Grundlagen Math. 31, 569-573 (1985; Zbl 0575.51016)] is generalized: finite Cayley-Klein geometries are defined, in which concepts as orthogonality, congruence and motions are declared. The points and lines of these geometries are the cuts of the lines and planes through a given point with a nondegenerate quadric of a finite projective space over a field \(K\) of odd order. It is shown that there are exactly nine Cayley-Klein geometries over \(K\), just as over the field of real numbers. Klein models and Poincaré models of these geometries are described.
Mathematics Subject Classification ID: 51M99
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 51F20
Mathematics Subject Classification ID: 03C65
zbMATH DE Number: 3984006
zbMATH Keywords: finite metric geometry
zbMATH Keywords: finite Cayley-Klein geometries
zbMATH Keywords: orthogonality
zbMATH Keywords: congruence
zbMATH Keywords: motions
zbMATH Keywords: Klein models
zbMATH Keywords: Poincaré models
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3984006