Characterization of harmoniously halfordered planes (Q1385223)

In a half-ordered plane, one has a betweenness function \(\alpha\) assigning a value in \(\{1,-1\}\) to any triplet \((a,b,c)\) of collinear points such that \(a\notin\{b,c\}\), and satisfying a cancellation rule as well as Pasch's Axiom. Every finite half-ordered plane is a semi-affine plane in the sense of \textit{P. Dembowski} [Arch. Math. 13, 120-131 (1962; Zbl 0135.39304)]. Let \(Z(a,b,c)\) denote the number of elements of \(\{(a,b,c),(b,c,a),(c,a,b)\}\) that are mapped to \(-1\). \textit{E. Sperner }[Math. Ann. 121, 107-130 (1949; Zbl 0032.17801)] has shown that the values of \(Z\) on a given geometry are either all even or all odd. In the latter case, the half-order is called `harmonious' (the English `harmonic' might have been a more suitable translation for the German `harmonisch'). In this case, one distinguishes three types of collinear quadruplets of points, according to the values that \(Z\) takes on the four triplets formed from the quadruplet. Nonexistence of quadruplets of type III is equivalent to an orientability criterion. In the paper under review, the following is proved for harmoniously half-ordered planes \((E,{\mathfrak G},\alpha)\) such that \((E,{\mathfrak G})\) is not the affine plane of order \(3\): (i) all quadruplets have type I iff \(\alpha\) is an order, (ii) not all quadruplets have type II, (iii) all quadruplets have type III iff \((E,{\mathfrak G})\) has order \(3\), (iv) there is no plane with quadruplets of type I and of type III, but none of type II, (v) there are quadruplets of type II and of type III but none of type I iff \((E,{\mathfrak G})\) is the affine plane of order \(7\), (vi) in every finite plane, except those in cases (iii) and (v), there are quadruplets of all three types, (vii) if \(E\) is infinite and \(\alpha\) is not an order then there are quadruplets of type I and of type II. Finally, the author constructs an example of an infinite plane containing quadruplets of types I and II but none of type III.