an:04036595
Zbl 0636.51012
Kist, G??nter
Theorie der verallgemeinerten kinematischen R??ume. (Theory of generalized kinematic spaces)
DE
Result. Math. 12, 325-347 (1987).
00153607
1987
j
51J05 20N05 51A15 51A45
metric plane; kinematic space; 2-set; incidence loop; ovoid; incidence groups; Moufang incidence loops with parallelism
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.
H.Hotje