An axiomatic foundation of Cayley-Klein geometries
The author gives an axiomatic characterization of the Cayley-Klein geometries with the exception of the doubly-hyperbolic plane, generalizing the characterization of plane absolute geometry introduced by \textit{F. Bachmann} [Aufbau der Geometrie aus dem Spiegelungsbegriff. 2. ergänzte Aufl. 2nd. ext. ed. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0254.50001)]. The axioms for the Cayley-Klein geometries are chosen so that the principle of duality holds. The basic features of the characterization are: Let \(G\) be a group and \(S\) and \(P\) invariant subsets of involutions in \(G\) such that \(S \cup P\) generates \(G\). Elements \(a, b, c, \ldots\) in \(S\) are called lines and elements \(A,B,C,\ldots\) in \(P\) are called points. The relation \(\alpha|\beta\) indicates that \(\alpha,\beta\) and \(\alpha\beta\) are involutions. A point \(A\) and a line \(b\) are incident if \(A|b\). The lines \(a\) and \(b\) are perpendicular if \(a|b\). The author stipulates a number of properties for the relation \(\alpha|\beta\). In the main part of the paper, the author establishes the desired characterization. In an appendix, the author provides a first order axiomatization of the above characterization.
scientific article; zbMATH DE number 6680043
scientific article