An order analysis of hyperfinite Borel equivalence relations

In this paper we first consider hyperfinite Borel equivalence relations with a pair of Borel Z-orderings. We define a notion of compatibility between such pairs, and prove a dichotomy theorem which characterizes exactly when a pair of Borel Z-orderings are compatible with each other. We show that, if a pair of Borel Z-orderings are incompatible, then a canonical incompatible pair of Borel Z-orderings of E₀ can be Borel embedded into the given pair. We then consider hyperfinite-over-finite equivalence relations, which are countable Borel equivalence relations admitting Borel Z²-orderings. We show that if a hyperfinite-over-hyperfinite equivalence relation E admits a Borel Z²-ordering which is compatible with itself, then E is hyperfinite.