Beyond Abstract Elementary Classes: On The Model Theory of Geometric Lattices

TL;DR

This paper explores the model theory of geometric lattices, identifying classes with stability properties and independence calculus, revealing both their stability and unusual features like the failure of the Smoothness Axiom.

Contribution

It introduces classes of geometric lattices with well-behaved stability theory, including an $ ext{ω}$-stable class with a monster model and non-forking independence, expanding beyond abstract elementary classes.

Findings

$( extbf{K}^3, rianglelefteq)$ is $ ext{ω}$-stable

It has a monster model and a non-forking independence calculus

The class fails the Smoothness Axiom, thus not an AEC

Abstract

Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, $(\mathbf{K}^3, \preccurlyeq)$, is $\omega$-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in $(\mathbf{K}^3, \preccurlyeq)$ the Smoothness Axiom fails, and so $(\mathbf{K}^3, \preccurlyeq)$ is not an $\mathrm{AEC}$.