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Borel homomorphisms of smooth -ideals

Given a countable Borel equivalence relation E on a Polish space, letIE denote the -ideal generated by the Borel partial transversals of E. We show that there is a Borel homomorphism fromIE toIF if and only if there is a smooth-to-one Borel homomorphism from a nite 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 hypernite. Using this, we prove that all 1 sets and 1 quasi-orders are Borel on Borel reducible to the quasi-order of Borel homomorphism on the class of inhomogeneous 1 on 1 -ideals.