On the Model Theory of Open Incidence Structures: The Rank 2 Case
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913593228787712 |
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| author | Paolini, Gianluca Quadrellaro, Davide Emilio |
| author_facet | Paolini, Gianluca Quadrellaro, Davide Emilio |
| contents | Taking inspiration from [1, 21, 24], we develop a general framework to deal with the model theory of open incidence structures. In this first paper we focus on the study of systems of points and lines (rank $2$). This has a number of applications, in particular we show that for any of the following classes all the non-degenerate free structures are elementarily equivalent, and their common theory is decidable, strictly stable, and with no prime model: $(k, n)$-Steiner systems (for $2 \leq k < n$); generalised $n$-gons (for $n \geq 3$); $k$-nets (for $k \geq 3$); affine planes; projective Möbius, Laguerre and Minkowski planes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Model Theory of Open Incidence Structures: The Rank 2 Case Paolini, Gianluca Quadrellaro, Davide Emilio Logic 03C45 03C65 03C98 05B30 51B05 51E05 Taking inspiration from [1, 21, 24], we develop a general framework to deal with the model theory of open incidence structures. In this first paper we focus on the study of systems of points and lines (rank $2$). This has a number of applications, in particular we show that for any of the following classes all the non-degenerate free structures are elementarily equivalent, and their common theory is decidable, strictly stable, and with no prime model: $(k, n)$-Steiner systems (for $2 \leq k < n$); generalised $n$-gons (for $n \geq 3$); $k$-nets (for $k \geq 3$); affine planes; projective Möbius, Laguerre and Minkowski planes. |
| title | On the Model Theory of Open Incidence Structures: The Rank 2 Case |
| topic | Logic 03C45 03C65 03C98 05B30 51B05 51E05 |
| url | https://arxiv.org/abs/2411.10792 |