Examples of non-tame abstract elementary classes of abelian groups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914545026465792 |
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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 an abstract elementary class $K_1$ of torsion-free abelian groups such that $K_1$ is not $(<\aleph_0)$-tame but is $\aleph_0$-tame. This answers a question of [BoVa17]. Furthermore, for every regular uncountable cardinal $μ$ less than the first measurable cardinal, we construct an abstract elementary class $K_2(2^μ)$ of torsion-free abelian groups such that $K_2(2^μ)$ is not $(<μ)$-tame.
$K_1$ and $K_2(2^μ)$ are non-tame for algebraic reasons. Furthermore, they constitute the first examples of non-tame abstract elementary classes in a natural language. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07893 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Examples of non-tame abstract elementary classes of abelian groups Herden, Daniel Mazari-Armida, Marcos Walton, Michael D. Logic Group Theory Primary: 03C60, 03C48. Secondary: 03C45, 20K20, 20K40 We construct an abstract elementary class $K_1$ of torsion-free abelian groups such that $K_1$ is not $(<\aleph_0)$-tame but is $\aleph_0$-tame. This answers a question of [BoVa17]. Furthermore, for every regular uncountable cardinal $μ$ less than the first measurable cardinal, we construct an abstract elementary class $K_2(2^μ)$ of torsion-free abelian groups such that $K_2(2^μ)$ is not $(<μ)$-tame. $K_1$ and $K_2(2^μ)$ are non-tame for algebraic reasons. Furthermore, they constitute the first examples of non-tame abstract elementary classes in a natural language. |
| title | Examples of non-tame abstract elementary classes of abelian groups |
| topic | Logic Group Theory Primary: 03C60, 03C48. Secondary: 03C45, 20K20, 20K40 |
| url | https://arxiv.org/abs/2605.07893 |