Ordered absolute geometry

Bachmann’s absolute geometry provides a common foundation of Euclidean and hyperbolic geometry without any assumptions about order or free mobility. An order structure can be introduced in an additional step either in a synthetic way (by postulating that every line is linearly orderable and admits a partition into sides) or analytically (by the requirement that the field of coordinates is orderable). The class of ordered absolute planes can be divided into two subclasses depending on whether or not the groups of rotations around a point O are cyclically ordered groups. In this paper we study the geometric properties of both kinds of order structures and determine the associated models. Surprisingly, in every ordered absolute plane the groups of rotations around a point O are cyclically orderable, with the only exception of subplanes of Euclidean planes with a negative orthogonality constant. We show that this exceptional case has a deeper reason: These Euclidean planes are subplanes of a Minkowskian plane and inherit the Minkowskian metric. The article concludes with some remarks on the foundations of ordered absolute geometry, which include comments on the well-known axiom systems of Hilbert and Tarski and on the role of the axiom of Pasch.