SB-property on metric structures

Camilo Argoty; Alexander Berenstein; Nicolas Cuervo Ovalle

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arXiv · 2302.01220

SB-property on metric structures

Camilo Argoty·Alexander Berenstein·Nicolas Cuervo Ovalle

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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.

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BibTeX RIS

Camilo Argoty, Alexander Berenstein, Nicolas Cuervo Ovalle. 2023-02-02. SB-property on metric structures. https://arxiv.org/abs/2302.01220

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