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AN ORDER ANALYSIS OF HYPERFINITE BOREL EQUIVALENCE RELATIONS
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 21 March 2025, pp. 1-23
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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.
