Weakly precomplete equivalence relations in the Ershov hierarchy

We study the computable reducibility $\leq_c$ for equivalence relations in the Ershov hierarchy. For an arbitrary notation $a$ for a nonzero computable ordinal, it is stated that there exist a $\Pi^{-1}_a$-universal equivalence relation and a weakly precomplete $\Sigma^{-1}_a$-universal equivalence relation. We prove that for any $\Sigma^{-1}_a$ equivalence relation $E$, there is a weakly precomplete $\Sigma^{-1}_a$ equivalence relation $F$ such that $E\leq_c F$. For finite levels $\Sigma^{-1}_m$ in the Ershov hierarchy at which $m=4k+1$ or $m=4k+2$, it is shown that there exist infinitely many $\leq_c$-degrees containing weakly precomplete, proper $\Sigma^{-1}_m$ equivalence relations.