Ordered groups and ordered geometries

The article studies the relationship between ordered groups and ordered geometries. To this end we consider metric planes of a very general kind (without any assumptions about order, continuity, free mobility and the existence or uniqueness of a joining line) which are singular (the set of translations forms a group) and show that the geometric concept of an ordered plane corresponds on the group-theoretical side to an order structure of the group of translations. This correspondence shows that not only Euclidean planes but also Minkowskian and Galilean planes are orderable if and only if the associated coordinate field is orderable. The article closes with some implications for the foundations of ordered geometry which include an axiomatic analysis of the Pasch axiom and some remarks on the relationship of the notions of incidence and order.