SB-property on metric structures

TL;DR

This paper explores the Schr"oder-Bernstein property in continuous theories, demonstrating its presence in certain structures like Hilbert spaces and probability algebras, and its absence in strictly stable theories.

Contribution

It extends the SB-property concept to continuous theories, introduces the SB-property up to perturbations, and analyzes its behavior in various continuous model-theoretic contexts.

Findings

Hilbert spaces with a bounded self-adjoint operator have the SB-property up to perturbations.

Atomless probability algebras with a generic automorphism have the SB-property up to perturbations.

Strictly stable theories do not have the SB-property in the continuous setting.

Abstract

A complete theory $T$ has the Schr\"oder-Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and any completion of the theory of probability algebras. We also study a weaker notion, the SB-property up to perturbations. This property holds if any two elementarily bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. We also study how the SB-property behaves with respect to randomizations. Finally we prove, in the continuous setting, that if $T$ is a strictly stable theory then $T$ does not have the SB-property.