When does elementary bi-embeddability imply isomorphism?

TL;DR

This paper characterizes when elementary bi-embeddability implies isomorphism in first-order theories, linking the Schroder-Bernstein property to classifiability and stability conditions.

Contribution

It proves that countable theories with the Schroder-Bernstein property are superstable, classifiable, and satisfy a specific nonmultidimensionality condition, providing new insights into their structure.

Findings

Theories with the Schroder-Bernstein property are superstable.

Such theories are classifiable with NDOP and NOTOP.

They satisfy a stronger nonmultidimensionality condition.

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 a countable theory T has the Schroder-Bernstein property then it is classifiable (it is superstable and has NDOP and NOTOP) and satisfies a slightly stronger condition than nonmultidimensionality, namely: there cannot be a model M of T, a type p over M, and an automorphism f of M such that for every two distinct natural numbers i and j, f^i(p) is orthogonal to f^j(p). We also make some conjectures about how the class of theories with the Schroder-Bernstein property can be characterized.