SB-property on metric structures
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929276776873984 |
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| author | Argoty, Camilo Berenstein, Alexander Ovalle, Nicolas Cuervo |
| author_facet | Argoty, Camilo Berenstein, Alexander Ovalle, Nicolas Cuervo |
| contents | A complete theory $T$ has the Schröder-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_01220 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | SB-property on metric structures Argoty, Camilo Berenstein, Alexander Ovalle, Nicolas Cuervo Logic 03C45, 03C66 A complete theory $T$ has the Schröder-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. |
| title | SB-property on metric structures |
| topic | Logic 03C45, 03C66 |
| url | https://arxiv.org/abs/2302.01220 |