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arXiv · 2605.07893

Examples of non-tame abstract elementary classes of abelian groups

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

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 $\mu$ less than the first measurable cardinal, we construct an abstract elementary class $K_2(2^\mu)$ of torsion-free abelian groups such that $K_2(2^\mu)$ is not $(<\mu)$-tame. $K_1$ and $K_2(2^\mu)$ are non-tame for algebraic reasons. Furthermore, they constitute the first examples of non-tame abstract elementary classes in a natural language.

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Daniel Herden, Marcos Mazari-Armida, Michael D. Walton. 2026-05-08. Examples of non-tame abstract elementary classes of abelian groups. https://arxiv.org/abs/2605.07893

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