Finite Cayley-Klein geometries.
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.
scientific article; zbMATH DE number 3984006
scientific article