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

Let $\le_{c}$ be computable reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceer $R$ with infinitely many equivalence classes, the index sets $\{i: R_{i}\le_{c} R\}$ (with $R$ non-universal), $\{i: R_{i}\ge_{c} R\}$, and $\{i: R_{i}\equiv_{c} R\}$ are $\Sigma^{0}_{3}$ complete, whereas in case $R$ has only finitely many equivalence classes, we have that $\{i: R_{i}\le_{c} R\}$ is $\Pi^{0}_{2}$ complete, and $\{i: R_{i}\ge_{c} R\}$ (with $R$ having at least two distinct equivalence classes) is $\Sigma^{0}_{2}$ complete. Next, solving an open problem from \cite{ceers}, we prove that the index set of the effectively inseparable ceers is $\Pi^{0}_{4}$ complete. Finally, we prove that the $1$-reducibility pre-ordering on c.e.~sets is a $\Sigma^{0}_{3}$ complete pre-ordering relation, a fact that is used to show that the pre-ordering relation $\le_{c}$ on ceers is a $\Sigma^{0}_{3}$ complete pre-ordering relation