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arXiv · 2604.03080

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

Daniel Herden·Marcos Mazari-Armida·Michael D. Walton
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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.

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Daniel Herden, Marcos Mazari-Armida, Michael D. Walton. 2026-04-03. An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example. https://arxiv.org/abs/2604.03080

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