Algebraic description of limit models in classes of abelian groups

TL;DR

This paper characterizes limit models in classes of abelian groups and torsion-free abelian groups, providing explicit algebraic structures and analyzing their properties within the framework of abstract elementary classes.

Contribution

It offers a precise algebraic description of limit models in these classes and establishes their model-theoretic properties such as stability and tameness.

Findings

Limit models in abelian groups are isomorphic to direct sums involving rationals and Prüfer groups.

Limit models in torsion-free abelian groups depend on chain cofinality, with explicit structures for uncountable cofinality.

Finitely Butler groups form an AEC that is Galois-stable and tame.

Abstract

We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show: $\textbf{Theorem}$ (1) If $G$ is a limit model of cardinality $\lambda$ in the class of abelian groups with the subgroup relation, then $G \cong (\oplus_{\lambda}\mathbb{Q}) \oplus \oplus_{p \text{ prime}} (\oplus_{\lambda} \mathbb{Z}(p^\infty))$. (2) If $G$ is a limit model of cardinality $\lambda$ in the class of torsion-free abelian groups with the pure subgroup relation, then: * If the length of the chain has uncountable cofinality, then $ G \cong (\oplus_{\lambda} \mathbb{Q} ) \oplus \Pi_{p\text{ prime}} \overline{(\oplus_{\lambda} \mathbb{Z}_{(p)})}$. * If the length of the chain has countable cofinality, then $G$ is not algebraically compact. We also study the class of finitely Butler groups with the pure subgroup relation, we show that it is an AEC, Galois-stable and $(<\aleph_0)$-tame and short.