Weakly precomplete equivalence relations in the Ershov hierarchy
We study the computable reducibility c 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-universal equivalence relation and a weakly precomplete X-1-universal equivalence relation. We prove that for any X-1 equivalence relation E, there is a weakly precomplete X-1 equivalence relation F such that E c F. For finite levels S1 in the Ershov hierarchy at which m = 4k + 1 or m = 4k + 2, it is shown that there exist infinitely many c-degrees containing weakly precomplete, proper Xm1 equivalence relations.