Abstract

In this paper we first consider hyperfinite Borel equivalence relations with a pair of Borel $\mathbb {Z}$ -orderings. We define a notion of compatibility between such pairs, and prove a dichotomy theorem which characterizes exactly when a pair of Borel $\mathbb {Z}$ -orderings are compatible with each other. We show that, if a pair of Borel $\mathbb {Z}$ -orderings are incompatible, then a canonical incompatible pair of Borel $\mathbb {Z}$ -orderings of $E_0$ can be Borel embedded into the given pair. We then consider hyperfinite-over-finite equivalence relations, which are countable Borel equivalence relations admitting Borel $\mathbb {Z}^2$ -orderings. We show that if a hyperfinite-over-hyperfinite equivalence relation E admits a Borel $\mathbb {Z}^2$ -ordering which is self-compatible, then E is hyperfinite.