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Approximate isomorphism of metric structures.

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Title: Approximate isomorphism of metric structures.
Authors: Hanson, James E.1 (AUTHOR) jhanson9@umd.edu
Source: Mathematical Logic Quarterly. Nov2023, Vol. 69 Issue 4, p482-507. 26p.
Subjects: Metric spaces, Approximate reasoning, Isomorphism (Mathematics)
Geographic Terms: Carlisle (Pa.)
Abstract: We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach‐Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two‐sorted structures that witness approximate isomorphism. As an application, we show that the theory of any R$\mathbb {R}$‐tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8]. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Logic Quarterly is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach‐Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two‐sorted structures that witness approximate isomorphism. As an application, we show that the theory of any R$\mathbb {R}$‐tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8]. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematical Logic Quarterly is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1002/malq.202200076
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        PageCount: 26
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    Subjects:
      – SubjectFull: Metric spaces
        Type: general
      – SubjectFull: Approximate reasoning
        Type: general
      – SubjectFull: Isomorphism (Mathematics)
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              Text: Nov2023
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