We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.

We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ11 (...) equivalence relations and show that existence of infinitely many properly Σ11 equivalence classes that are complete as Σ11 sets is necessary but not sufficient for a relation to be complete in the context of Σ11 equivalence relations. (shrink)