Recent developments on absolute geometries and algebraization by K-loops
A spatial version of Pasch's assertion is obtained by starting from an ordered space \((P,{\mathcal L},\alpha),\) where \(P\) is the set of points, \({\mathcal L}\) the set of lines and \(\alpha \) the betweenness relation. It allows to define the concepts \(h\)-parallel, one sided h-parallel and hyperbolic incidence structure. A characterization of spaces with congruence \((P,{\mathcal L},\equiv)\) admitting point reflections is presented. An absolute space \((P,{\mathcal L},\alpha ,\equiv)\) is obtained by joining together incidence, congruence and order structures assumed to satisfy some compatibility conditions. A \(K\)-loop \((P,+)\) and an incidence fibration in the sense of \textit{E. Zizioli} [J. Geom. 30, 144-156 (1987; Zbl 0632.51019)] are associated to an absolute space satisfying certain conditions. They can be used for an algebraization of these spaces.
scientific article; zbMATH DE number 1369794