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

TL;DR

This paper develops a model-theoretic framework for open incidence structures, focusing on rank 2 systems like Steiner systems, generalized polygons, and affine and projective planes, showing their theories are decidable, stable, and elementarily equivalent.

Contribution

It introduces a general framework for the model theory of open incidence structures and proves properties for various rank 2 classes, including decidability and stability.

Findings

All non-degenerate free structures in the studied classes are elementarily equivalent.

Theories of these structures are decidable and strictly stable.

These structures have no prime models.

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.