Definable coordinate geometries over fields

We define general notions of coordinate geometries over fields and ordered fields, and consider coordinate geometries that are given by finitely many relations that are definable over those fields. We show that the automorphism group of such a geometry determines the geometry up to definitional equivalence; moreover, if we are given two such geometries G and G′, then the concepts (explicitly definable relations) of G are concepts of G′ exactly if the automorphisms of G′ are automorphisms of G. We show this by first proving that a relation is a concept of G exactly if it is closed under the automorphisms of G and is definable over the field; moreover, it is enough to consider automorphisms that are affine transformations.

We show how this result can be applied to quickly determine relationships and differences between various geometries and spacetimes, including ordered affine, Euclidean, Galilean, Newtonian, Late Classical, Relativistic and Minkowski spacetimes (we first define these spacetimes and geometries using a Tarskian first-order language centred on the ternary relation Bw of betweenness). We conclude with a selection of open problems related to the existence of certain intermediate geometries.