Ordered metric geometry

Abstract

Metric geometry in the sense of Hjelmslev and Bachmann studies metric planes of a very general kind without any assumption about order, continuity and the existence and uniqueness of joining lines. An order structure can be defined in an additional step by introducing a relation of betweenness which satisfies the axioms of order of Hilbert’s Grundlagen der Geometrie, i.e., one-dimensional axioms which characterize the linear order of collinear points and a single plane order axiom which was proposed by Pasch. The Pasch axiom however is based on the assumption that any two points have a unique joining line. This is not necessarily satisfied by Cayley–Klein geometries (e.g. by Minkowskian planes) and even in plane absolute geometry the Pasch axiom is not a necessary condition for an ordering of the associated field of coordinates (see Sect. 5). The aim of this article is to introduce an order structure for the widest class of metric planes (without any assumption about the existence of joining lines, free mobility or some form of a parallel axiom) and to show that the correspondence between geometrical and algebraical order structures, which is well-known in affine and projective geometry, can be extended to plane absolute geometry. The article closes with a discussion of the role of the Pasch axiom in ordered metric geometry. An axiomatization of ordered metric planes in a first-order language is provided in an Appendix.