An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example

TL;DR

This paper constructs a specific class of torsion-free abelian groups forming an abstract elementary class with unique stability and amalgamation properties, highlighting core mechanisms of prior constructions.

Contribution

It introduces a variation of Paolini-Shelah's example that isolates key mechanisms, demonstrating a class with particular model-theoretic properties.

Findings

The class is not stable.

It has the joint embedding property and no maximal models.

It lacks the amalgamation property but is (<ℵ₀)-tame.

Abstract

We construct a class $\hat{K}$ of torsion-free abelian groups such that $\hat{\mathbf{K}}=(\hat{K}, \leq_p)$ is an abstract elementary class with $\operatorname{LS}(\hat{\mathbf{K}})=\aleph_0$ such that: $(\cdot)$ $\hat{\mathbf{K}}$ is not stable; $(\cdot)$ $\hat{\mathbf{K}}$ has the joint embedding property and no maximal models, but does not have the amalgamation property; $(\cdot)$ $\hat{\mathbf{K}}$ is $(<\aleph_0)$-tame. The class we construct is a variation of [PaSh, Section 4] which isolates the core mechanism of the Paolini-Shelah construction.