References & Citations
Mathematics > Logic
Title: An order analysis of hyperfinite Borel equivalence relations
(Submitted on 26 Apr 2024 (v1), last revised 7 May 2024 (this version, v2))
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.
Submission history
From: Ming Xiao [view email][v1] Fri, 26 Apr 2024 16:36:06 GMT (18kb)
[v2] Tue, 7 May 2024 02:36:44 GMT (24kb)
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