Definable coordinate geometries over fields, part 1: theory

TL;DR

This paper develops a theoretical framework for coordinate geometries over fields, showing how automorphism groups uniquely determine the geometry and its definable relations, with implications for understanding geometric structures through automorphisms.

Contribution

It introduces a general theory linking automorphism groups to coordinate geometries over fields, establishing conditions for definability and equivalence.

Findings

Automorphism groups determine the geometry up to definitional equivalence.

Relations definable over the field are exactly those closed under automorphisms.

Affine transformations suffice to analyze automorphisms in these geometries.

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 $\mathcal{G}$ and $\mathcal{G}'$, then the concepts (explicitly definable relations) of $\mathcal{G}$ are concepts of $\mathcal{G}'$ exactly if the automorphisms of $\mathcal{G}'$ are automorphisms of $\mathcal{G}$. We show this by first proving that a relation is a concept of $\mathcal{G}$ exactly if it is closed under the automorphisms of $\mathcal{G}$ and is definable over the field; moreover, it is enough to consider automorphisms that are affine transformations.