Vectorspacelike representation of absolute planes (Q884676)

Let \((\mathbf{E},\mathcal{G},\alpha,\equiv)\) be an absolute plane, where \(\mathbf{E}\) and \(\mathcal{G}\) are the set of points and lines, respectively, \(\alpha\) is an order function, and \(\equiv\) is a congruence relation [see \textit{H. Karzel, K. Sörensen} and \textit{D. Windelberg}, Einführung in die Geometrie, Vandenhoeck, Göttingen (1973; Zbl 0248.50001)] with a multiplication defined on the points. The authors investigate a great number of properties of the plane. They show e.g., if \((\mathbf{E},\mathcal{G},\alpha,\equiv)\) is singular, then \((\mathbf{E},+)\) is a commutative group; also \((\mathbf{E},\mathcal{G},\alpha,\equiv)\) is vectorspacelike if it is Euclidean.