Regular incidence permutation sets and incidence quasigroups

An incidence group (P,R·) can be considered as an incidence space (P,R) together with a groupC(=P set of left translations) of collineations of (P,R) acting regularly and fixed point free onP. Here we replaceC by a regular collineation set with fixed points. Then (P,R,C) can be turned only in an incidence quasigroup. Examples of such structures can be derived from arbitrary absolute spaces (P,R,). If denotes the set of all reflectionsp in pointsp ofP then is a set of collineations, even of dilatations, of the incidence space (P,R) acting regularly on the setP of points. Therefore we will study more precisely the case whereC consists of weak dilatations (§2) and then apply the results on geometric structures like kinematic spaces, affine geometries (§5) and hyperbolic planes (§6). By that we present new properties of these generalized dilatations, which were introduced by G. Kist and B. Reinmidl.This research was partially supported by M.U.R.S.T. (40% grant) and by G.N.S.A.G.A. of Italian C.N.R. while the first Author was Visiting Professor under a C.N.R. grant.Dedicated to Günter Pickert on the occasion of his 80th birthday