Suppose $E$ and $F$ are Borel equivalence relations on Polish spaces $X$, $Y$, resp. Say that $E$ is **surjectively Borel reducible** to $F$ iff there is a Borel surjection $f:X \to Y$ such that $xEy$ iff $f(x) F f(y)$. This is (at least on the surface) stronger than normal Borel reducibility, weaker than Borel isomorphism.

What do we know about this reducibility? If $E$ is effectively Borel with only countably many classes, is $E$ effectively surjectively reducible to the trivial equivalence relation on $\omega$?

edit: Probably this is the version I am interested in: Say $E$ is **strongly Borel reducible** to $F$ iff $E$ reduces to $F$ via a Borel $f$ such that the range of $f$ is Borel. When $Y=\omega$ and $E$ has infinitely many classes, "$E$ is effectively stongly Borel reducible to the relation $F$ on $Y$" exactly means "$E$ is effectively surjectively Borel reducible to $F$".

edit again: Actually, the arXiv version of Fokina-Friedman-T¨ornquist "The Effective Theory of Borel Equivalence Relations" probably has a (minor) mistake. On page 9, the equivalence relation $E^*$ is $\Pi^1_1$, not Hyp. The $\Sigma^1_1$ line

$$(\exists x_0,x_1) ( x_0 \in H(d_0) \wedge x_1 \in H(d_1) \wedge x_0 E x_1)$$
fails when $H(d_0) = \emptyset$. The proof of Proposition 2 can be easily repaired, but the issue of empty set is worth attention when we seriously require the range being $\Delta^1_1$.