DEFINABLE COORDINATE GEOMETRIES OVER FIELDS

Abstract 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 $\mathcal {G}$ script upper G and G ′ $\mathcal {G}'$ script upper G prime , then the concepts (explicitly definable relations) of G $\mathcal {G}$ script upper G are concepts of G ′ $\mathcal {G}'$ script upper G prime exactly if the automorphisms of G ′ $\mathcal {G}'$ script upper G prime are automorphisms of G $\mathcal {G}$ script upper G . We show this by first proving that a relation is a concept of G $\mathcal {G}$ script upper G exactly if it is closed under the automorphisms of G $\mathcal {G}$ script upper G