Examples of non-tame abstract elementary classes of abelian groups

TL;DR

The paper constructs specific examples of abstract elementary classes of torsion-free abelian groups that demonstrate non-tameness properties, answering an open question and providing the first natural language examples.

Contribution

It introduces the first natural language examples of non-tame abstract elementary classes and addresses an open question about tameness in such classes.

Findings

Constructed an AEC of torsion-free abelian groups that is not (<aleph_0)-tame but is aleph_0-tame.

Constructed AECs of torsion-free abelian groups that are not (<mu)-tame for certain uncountable cardinals.

Provided the first natural language examples of non-tame AECs.

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.