On the Model Theory of Open Incidence Structures: The Rank 2 Case

Gianluca Paolini; Davide Emilio Quadrellaro

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arXiv · 2411.10792

On the Model Theory of Open Incidence Structures: The Rank 2 Case

Gianluca Paolini·Davide Emilio Quadrellaro

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Abstract

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 M\"obius, Laguerre and Minkowski planes.

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BibTeX RIS

Gianluca Paolini, Davide Emilio Quadrellaro. 2024-11-16. On the Model Theory of Open Incidence Structures: The Rank 2 Case. https://arxiv.org/abs/2411.10792

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