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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.10279 |
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| _version_ | 1866916842944069632 |
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| author | Madarász, Judit Stannett, Mike Székely, Gergely |
| author_facet | Madarász, Judit Stannett, Mike Székely, Gergely |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10279 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Definable coordinate geometries over fields, part 1: theory Madarász, Judit Stannett, Mike Székely, Gergely Logic 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. |
| title | Definable coordinate geometries over fields, part 1: theory |
| topic | Logic |
| url | https://arxiv.org/abs/2507.10279 |