Stability of the non-positive-syntactic relation

Determine whether the explicitly constructed relation \(\preccurlyeq\in\mathscr{L}^{1}_{\mathbb{Z}}\), defined by preservation of divisibility by every positive integer for infinite-order elements, yields a stable abstract elementary class \((\mathbb{Z},\preccurlyeq)\).

Background

Theorem \ref{n-syntac} constructs a strong submodel relation below purity that is not positive syntactic. The relation requires that, for every infinite-order element of the smaller group and every positive integer nn, divisibility by nn agree between the smaller and larger groups.

The paper shows that this relation does not have the amalgamation property, but does not establish whether the corresponding AEC is stable.

References

The relation from Theorem~\ref{n-syntac} does not have the amalgamation property. The span p(\mathbb{Z}/p2\mathbb{Z})\preccurlyeq \mathbb{Z}/p2\mathbb{Z}, \quad p(\mathbb{Z}/p2\mathbb{Z})\preccurlyeq p(\mathbb{Z}/p2\mathbb{Z})\oplus\mathbb{Z} cannot be completed to a commutative square of strong embeddings. We do not know whether (\mathbb{Z},\preccurlyeq) is stable.

— The lattice of abstract elementary classes of modules  (2608.19548 - Hyttinen et al., 20 Aug 2026) in Section 5.1, Remark following Theorem \ref{n-syntac}