Hjelmslevgruppen mit Nachbar-Homomorphismus. (Hjelmslev groups with neighbour-homomorphism) (Q920393)

Hjelmslev groups are generalized metric planes as introduced by \textit{F. Bachmann} [Aufbau der Geometrie aus dem Spiegelungsbegriff. 2. Aufl. (1973; Zbl 0254.50001)], in which two points do not necessarily determine a line. Let \((G,S)\) and \((\bar G,\bar S)\) be Hjelmslev groups and \(E(G,S),\) \(E(\bar G,\bar S)\) be the group planes associated with \((G,S)\) and \((\bar G,\bar S)\) respectively. If \(P\) \((\bar P)\) denotes the set of points of \(E(G,S)\) \((E(\bar G,\bar S))\) then the homomorphism group \(f: G\to \bar G\) satisfies Sf\(\subseteq\bar S\), \(P_ f\subseteq \bar P\) is called Hjelmslev homomorphism from \((G,S)\) to \((\bar G,\bar S)\). The following requirements yield that \(E(G,S)\) can be embedded into the projective Hjelmslev plane over a local ring R. a) Let \(A,B\in P\) such that \(Af\neq Bf\) then there exists a unique line from S joining them. b) Each line from \(\bar S\) contains at least four points. Moreover G is isomorphic to a subgroup of an orthogonal group \(O^+_ 3(V,g).\) It is a very nice generalization of the main Bachmann theorem [loc. cit.].