On Σ11 equivalence relations over the natural numbers

We study the structure of Σ1 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 Σ1 (i.e. Σ 1 1 but not ∆ 1 1) equivalence classes. We also show the existence of incomparable Σ1 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ1 equivalence relations (under both reducibilities) and show that existence of infinitely many properly Σ1 equivalence classes that are complete as Σ 1 1 sets (under the corresponding reducibility on sets) is necessary but not sufficient for a relation to be complete in the context of Σ1 equivalence relations.