Archimedeisation of some ordered geometric structures which are related to kinematic spaces
\textit{D. Gröger} has studied [J. Geom. 25, 49-76 (1985; Zbl 0576.51012); Mitt. Math. Ges. Hamb. 11, 441-457 (1987; Zbl 0632.51015)] the problem of `archimedeising' absolute planes, i.e. the conditions that a non- archimedean ordered absolute plane has to satisfy such that one can `extract' from it in a natural way an archimedean ordered absolute plane, once a `unit distance' has been selected. Given the close relationship between absolute planes and kinematic spaces (as defined by the first author [Sympos. Math. 11, Algebra commut., Geometria, Convegni 1971/1972, 413-439 (1973; Zbl 0284.50017)]), the authors aim to `archimedeise' ordered kinematic spaces and lay, in this paper, the algebraic groundwork for this task. It leads through the study of linearly ordered groups, cyclic ordered groups, betweenness groups, and separated groups, to the definition of a notion of `archimedeanity' and `archimedeisation' for separated groups, and eventually to the complete discussion of the archimedeisation of three (out of four possible) types of separated groups that are derived from unitary associative algebras (A,K) of degree 2 over non-archimedean ordered fields K. The fourth type will be treated in another paper.
scientific article; zbMATH DE number 4202330