Lower levels of Euclidean planes (Q1267345)

A pre-Euclidean plane \(E\) is a plane where there exists a fixed-point free binary relation on the set of its lines. For each line \(l\) on \(E\) one can define a transformation \(f_l:E \rightarrow E\) such that only the points of \(l\) are fixed by \(f_l\), every line parallel to \(l\) is transformed into a line parallel to \(l\), for every point \(P \notin l\), the line defined by \(P\) and its image \(f_l(P)\) is in the binary relation with \(l\) and for every point \(R \in Pf_l(P)-l\) and every point \(Q \in l-Pf_l(P)\) the parallelogram that contains \(P\), \(f_l(P)\) and \(R\) as vertices shares a vertex with the parallelogram that includes \(P\), \(f_l(P)\) and \(Q\) among its vertices. Such a transformation \(f_l\) is called a right-reflection on \(E\). Analogously, one can define a left-reflection on \(E\). Let \(S\) be the group generated by the set of right-reflections and left-reflection on \(E\) and \(I\) the set of involutions on \(E\). One can give a geometric structure to \(S\) such that it becomes a pre-Euclidean plane. A pre-Euclidean plane is called reflection geometrically representable (RGR-plane) if there exist bijections between the set of lines respectively \(E\) and \(S\) that preserve incidence and the binary relations is \(l\) itself. The authors shows that a pre-Euclidean plane \(E\) is a RGR-plane if and only if the incidence structure on \(E\) is a translation plane with characteristic \(\neq 2\). He then characterizes the RGR-planes where every pre-reflection is a collineation and those where every pre-reflection is an orthogonal collineation, where the meaning of orthogonal is generalized to agree with the binary relation considered.