Theorie der verallgemeinerten kinematischen Räume. (Theory of generalized kinematic spaces)
Incidence spaces (G,\({\mathfrak G})\) are considered with regular sets of collineations, the so-called incidence loops. To any metric plane there belongs such a structure namely a kinematic space which is embedded into a projective space P such that the complement Q is a 2-set: \(Q\subset P\) is called a 2-set if for any line \(X\subset P\) one has \(| X\cap Q| \leq 2\) or \(X\subset Q\). In the first part of the paper a survey is given on those 2-sets, especially on s-homogeneous ones. Q is called regular, if \(P\setminus Q\) is an incidence loop. It is shown that knot ovals Q with \(| Q| \neq 4\) are not regular and that an ovoid in a 3-dimensional projective space can not be a 2-set of an incidence group. So, there is the existence problem: Does any regular Q contain at least one line? In the second part properties of incidence groups are generalized to incidence loops. Of special interest are the Moufang incidence loops with parallelism. Furthermore it is shown that any linear fibered Moufang incidence loop with a 2-set as a complement in a projective space is two- sided, and a classification of all those incidence loops is given.
Moufang incidence loops with parallelism
scientific article; zbMATH DE number 4036595
scientific article