On the Degree Structure of Equivalence Relations Under Computable Reducibility

Abstract

We study the degree structure of the $\omega$-c.e., $n$-c.e., and ${\Pi }_1^0$ equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the $\omega$-c.e. and $n$-computably enumerable equivalence relations. We provide computable enumerations of the degrees of $\omega$-c.e., $n$-c.e., and ${\Pi }_1^0$ equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.