Structurable equivalence relations

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.

arXiv comment77 pages; answered Remark 5.28 (Problem 9.2)