The complexity of index sets of classes of computably enumerable equivalence relations

Let ≤c be computable the reducibility on computably enumerable equivalence relations (or ceers).We show that for every ceer R with infinitely many equivalence classes, the index sets {i: Ri ≤c R} (with R nonuniversal), {i: Ri ≥c R}, and {i: Ri ≡c R} are Σ03 complete, whereas in case R has only finitely many equivalence classes, we have that {i: Ri ≤c R} is Π02 complete, and {i: Ri ≥c R} (with R having at least two distinct equivalence classes) is Σ02 complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is Π04 complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a Σ03 complete preordering relation, a fact that is used to show that the preordering relation ≤c on ceers is a Σ03 complete preordering relation.