A new construction for Minkowski planes (Q1381328)

The author constructs Minkowski planes over a pseudo-ordered field \(F\) from the group PGL(2,\(F\)) and two order preserving permutations of \(F\). The title of the paper, however, is misleading because the construction is based on the author's earlier paper [Zesz. Nauk. Geom. 20, 13-21 (1993; Zbl 0807.51005)] and the more group theoretic setting presented here was given by the reviewer [Beitr. Algebra Geom. 37, 355-366 (1996; Zbl 0873.51004)] including a determination of isomorphism classes and automorphisms not dealt with in the paper under review. The construction is illustrated with several examples for which certain groups of central automorphisms are given and the Klein-Kroll types of these Minkowski planes are determined. In the final section of the paper, the author, following \textit{H. A. Wilbrink} [Geom. Dedicata 12, 119-129 (1982; Zbl 0478.51005)] for finite Minkowski planes, associates with each point \(Z\) of a Minkowski plane over a pseudo-ordered field as constructed here an incidence structure consisting of all points not on a generator through \(Z\) and as line set the collection of all traces of generators or circles not passing through \(Z\) where one adds in the case of circles the point of intersection of the two generators through the deleted points not passing through \(Z\). It is shown that this incidence structure is a near-affine plane.