Halbgeordnete Ebenen. (On halfordered planes) (Q1118164)

Continuing his study on halforderings of arbitrary planes (linear spaces), based on Sperner's notion of betweenness functions, the author addresses the problem of the existence of nontrivial halforderings in special planes. He shows that under rather weak assumptions any halfordering not fulfilling the axiom (A3), which reads that there exist collinear points x,y,u,v with u between and v not between x and y, has the form \((a| b,c)=f(b)\cdot f(c)\) with a function f from the point set into \(\{\) 1,- 1\(\}\), and gives criteria for linear spaces in which all nontrivial halforderings fulfil (A3). He shows that any halfordering of an affine plane extends to a halfordering of the by one point extended affine plane, and that any halfordering of a by one point reduced projective plane extends to a halfordering of the projective plane. Studying the reverse constructions, he gives conditions under which the axiom (A3) remains valid, and which, in particular, hold in the desarguesian case. Since, by an earlier work of the author, any finite halfordered linear space with (A3) equals a desarguesian affine or projective plane, a by one point extended desarguesian affine, or a by one point reduced desarguesian projective plane, his results yield a complete overview over the halforderings of finite linear spaces.