The Schroder-Bernstein property for theories of abelian groups

TL;DR

This paper characterizes when the theory of an abelian group has the Schroder-Bernstein property, linking it to stability conditions and structural decompositions of the group.

Contribution

It establishes an equivalence between the Schroder-Bernstein property, omega-stability, and a specific structural form of abelian groups.

Findings

Th(G, +) has the Schroder-Bernstein property iff G is a sum of divisible and bounded torsion groups.

The property is equivalent to omega-stability of Th(G, +).

In saturated extensions, automorphisms are unipotent under certain conditions.

Abstract

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the Schroder-Bernstein property; 2. Th(G, +) is omega-stable; 3. G is the direct sum of a divisible group and a torsion group of bounded exponent; 4. Th(G, +) is superstable, and if (H, +) is a saturated elementary extension of (G,+), every map in Aut(H/H^0) is unipotent.