Weakly precomplete equivalence relations in the Ershov hierarchy
We study the computable reducibility <<sub>c</sub> for equivalence relations in the Ershov hierarchy. For an arbitrary notation a for a nonzero computable ordinal, it is stated that there exist а П<sup>-</sup>universal equivalence relation and a weakly precomplete X<sup>-1</sup>-universal equivalence relation. We prove that for any X<sup>-1</sup> equivalence relation E, there is a weakly precomplete X<sup>-1</sup> equivalence relation F such that E <<sub>c</sub> F. For finite levels S<sup>1</sup> in the Ershov hierarchy at which m = 4k + 1 or m = 4k + 2, it is shown that there exist infinitely many <<sub>c</sub>-degrees containing weakly precomplete, proper Xm<sup>1</sup> equivalence relations.