Structurable equivalence relations

TL;DR

This paper explores the structure and classification of countable Borel equivalence relations that can be endowed with structures from a class , revealing their lattice organization and connections to model-theoretic properties.

Contribution

It introduces the concept of -structurable equivalence relations, analyzes their interactions with Borel homomorphisms, and characterizes classes where all such relations are smooth.

Findings

The classes of -structurable equivalence relations form a distributive lattice.

The Borel reducibility among these relations contains a large sublattice.

Characterization of  classes where all -structurable relations are smooth.

Abstract

For a class $\mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there is a Borel way to put a structure in $\mathcal K$ on each $E$-equivalence class. We study in this paper the global structure of the classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$. We show that $\mathcal K$-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$, under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on $\mathcal K$-structurability of various model-theoretic properties of $\mathcal K$. In particular, we characterize the $\mathcal K$ such that every $\mathcal K$-structurable equivalence relation is smooth, answering a question of Marks.