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SB-property on metric structures

Camilo Argoty, Alexander Berenstein, Nicolas Cuervo Ovalle

Abstract

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.

SB-property on metric structures

Abstract

A complete theory 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 is a strictly stable theory then does not have the SB-property.
Paper Structure (6 sections, 19 theorems, 53 equations)

This paper contains 6 sections, 19 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Hilbert spaces and expansions with bounded self-adjoint operators.
  3. Expansions with self adjoint operators
  4. Probability algebras and expansions by a generic automorphism.
  5. Randomizations.
  6. Superstability and the SB-property

Key Result

Proposition 2.1

$IHS$ has the SB-property.

Theorems & Definitions (77)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.5
  • Definition 2.6
  • Corollary 2.8
  • proof
  • ...and 67 more