Archimedisierung absoluter Ebenen. (Construction of archimedean absolute planes)
The author considers an ordered absolute plane in the sense of \textit{H. Karzel}, \textit{K. Sörensen} and \textit{D. Windelberg}: Einführung in die Geometrie (1973; Zbl 0248.50001). Depending on the choice of a unit length e one has a distance measure d with values in a commutative ordered group. Select a point 0. The set of points x with \(d(x,0)<n e\) for some \(n\in {\mathbb{N}}\) admits a partition into classes: two points x,y belong to the same class if \(d(x,y)<n^{-1}e\) for every \(n\in {\mathbb{N}}\). These classes become the points of a new plane whose structure stems from the original one. The new plane is not always an absolute plane. Examples are given. Remark. Similar questions without taking ordering into account were studied e.g. by \textit{A. Dress} in Math. Z. 85, 116-140 (1964; Zbl 0124.131) and Abh. Math. Semin. Univ. Hamb. 32, 52-54 (1963; Zbl 0159.051).
scientific article; zbMATH DE number 3921373
scientific article