Charakterisierung harmonisch halbgeordneter ebenen

A halfordered plane is a plane with a betweenness function a, which satisfies the axioms (A1) and (A2) (Axiom of Pasch). A halfodered plane is called harmonious if for three collinear points a,b,c the number of the triplets (a,b,c), (b,c,a) and (c,a,b), which have the value -1 under the function a, is odd. For four collinear points of a harmonious halfodered plane one can distinguish three possible typs relative to the betweenness function α (cf. (2.4)). In this note we classify the halfodered planes by the appearance of this types. In a semiaffine plane of order 3 we have only points of Type III, in a affine plane of order 7 only points of type II and III. In all others finite semiaffine planes we have points of all three types. Furthermore there exist projective planes with points only of Type I and II. The main results are (3.8) and (4.3).