Borel homomorphisms of smooth σ-ideals
نویسندگان
چکیده
Given a countable Borel equivalence relation E on a Polish space, let IE denote the σ-ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism from IE to IF if and only if there is a smooth-to-one Borel homomorphism from a finite index Borel subequivalence relation of E to F . As a corollary, we see that IE is homogeneous in the sense of Zapletal (2007, Forcing Idealized, Preprint) if and only if E is hyperfinite. Using this, we prove that all Σ2 sets and Σ1 quasi-orders are Borel on Borel reducible to the quasi-order of Borel homomorphism on the class of inhomogeneous Π1 on Σ 1 1 σ-ideals.
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