The Schroder-Bernstein property for a-saturated models

TL;DR

This paper characterizes when a first-order theory's models exhibit the Schröder-Bernstein property, linking it to super stability, non-multidimensionality, and the absence of nomadic types.

Contribution

It establishes a precise criterion for the SB property in theories and their a-saturated models, connecting model-theoretic stability conditions with the SB property.

Findings

A superstable, non-multidimensional theory can be expanded to have the SB property.

Among superstable theories, the SB property for a-saturated models is equivalent to having no nomadic types.

The SB property is characterized by specific stability and dimensionality conditions.

Abstract

A first-order theory T has the Schr\"oder-Bernstein (SB) property if any pair of elementarily bi-embeddable models are isomorphic. We prove that T has an expansion by constants that has the SB property if and only if T is superstable and non-multidimensional. We also prove that among superstable theories T, the class of a-saturated models of T has the SB property if and only if T has no nomadic types.