The Schr\"oder-Bernstein property for weakly minimal theories

TL;DR

This paper characterizes when countable weakly minimal theories have the Schroeder-Bernstein property, linking it to orbit conditions and the absence of infinite bi-embeddable, nonisomorphic models, and shows its absoluteness in ZFC.

Contribution

It establishes an equivalence between the Schroeder-Bernstein property and specific orbit and model conditions for countable weakly minimal theories, and proves its absoluteness in ZFC.

Findings

Schroeder-Bernstein property characterized by orbit conditions.

Equivalence between bi-embeddability and isomorphism in these theories.

Property is absolute between transitive models of ZFC.

Abstract

For a countable, weakly minimal theory, we show that the Schroeder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to both a condition on orbits of rank 1 types and the property that the theory has no infinite collection of pairwise bi-embeddable, pairwise nonisomorphic models. We conclude that for countable weakly minimal theories, the Schroeder-Bernstein property is absolute between transitive models of ZFC.