An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917419822350336 |
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| author | Herden, Daniel Mazari-Armida, Marcos Walton, Michael D. |
| author_facet | Herden, Daniel Mazari-Armida, Marcos Walton, Michael D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03080 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example Herden, Daniel Mazari-Armida, Marcos Walton, Michael D. Logic Group Theory Primary: 03C60, 03C48. Secondary: 03C45, 20K20, 20K40 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. |
| title | An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example |
| topic | Logic Group Theory Primary: 03C60, 03C48. Secondary: 03C45, 20K20, 20K40 |
| url | https://arxiv.org/abs/2604.03080 |