Model Equivalences

TL;DR

This paper studies equivalence relations on models of a theory, called MERs, focusing on their definability, classification, and preservation properties in classical and continuous logic, revealing new hierarchies and relationships.

Contribution

It introduces and classifies definable MERs, explores their preservation properties, and uncovers intrinsic roles of continuous logic and interpretations in their structure.

Findings

Characterization of definable MERs in classical and continuous logic

Identification of hierarchy related to preservation of reducts

Results on MERs in stable theories

Abstract

We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, in a language appropriate for pairs of models. We provide many examples of definable MERs, along with the first steps of a classification theory for them. We characterize the special classes of definable MERs associated with preservation of formulas, either in classical first order logic or in continuous logic, and uncover an intrinsic role for the latter. We bring out a nontrivial relationship with interpretations (imaginary sorts), leading to a wider hierarchy of classes related to the preservation of reducts. We give results about the relationship between these classes, both for general theories and for theories satisfying additional model-theoretic properties, such as stability.