Simultaneous Reducibility of Pairs of Borel Equivalence Relations

TL;DR

This paper classifies pairs of smooth countable Borel equivalence relations based on their reducibility and isomorphism properties, extending previous notions and identifying key subclasses with complete invariants.

Contribution

It provides a complete classification of pairs of smooth countable Borel equivalence relations up to simultaneous Borel bireducibility and biembeddability, and generalizes Borel parametrization.

Findings

Classified all pairs of smooth countable Borel equivalence relations up to simultaneous Borel bireducibility.

Identified large subclasses where natural invariants are complete.

Presented counterexamples outside these subclasses.

Abstract

Let $E\subseteq F$ and $E'\subseteq F'$ be Borel equivalence relations on the standard Borel spaces $X$ and $Y$, respectively. The pair $(E,F)$ is simultaneously Borel reducible to the pair $(E',F')$ if there is a Borel function $f:X\to Y$ that is both a reduction from $E$ to $E'$ and a reduction from $F$ to $F'$. Simultaneous Borel embeddings and isomorphisms are defined analogously. We classify all pairs $E\subseteq F$ of smooth countable Borel equivalence relations up to simultaneous Borel bireducibility and biembeddability, and a significant portion of such pairs up to simultaneous Borel isomorphism. We generalize Mauldin's notion of Borel parametrization in order to identify large natural subclasses of pairs of smooth countable equivalence relations and of singleton smooth (not necessarily countable) equivalence relations for which the natural combinatorial isomorphism invariants are complete, and we present counterexamples outside these subclasses. Finally, we relate isomorphism of smooth equivalence relations and of pairs of smooth countable equivalence relations to Borel equivalence of Borel functions as discussed in Komisarski, Michalewski, and Milewski.