An embedding theorem of $\mathbb{E}_{0}$ with model theoretic applications

TL;DR

This paper introduces a new criterion for embedding the equivalence relation $E_0$ and applies it to model-theoretic equivalence relations, generalizing previous results on Borel cardinality and pseudo $F_\sigma$ groups.

Contribution

It provides a novel embedding criterion for $E_0$ and extends existing results in model theory regarding Borel complexity and group structures.

Findings

Established a new criterion for embedding $E_0$.

Generalized previous results on Borel cardinality of Lascar strong types.

Extended results on pseudo $F_\sigma$ groups.

Abstract

We provide a new criterion for embedding $\mathbb{E}_{0}$, and apply it to equivalence relations in model theory. This generalize the results of the authors and Pierre Simon on the Borel cardinality of Lascar strong types equality, and Newelski's results about pseudo $F_{\sigma}$ groups.