On the Degree Structure of Equivalence Relations Under Computable Reducibility

We study the degree structure of the ω-c.e., n-c.e. and Π1 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 ω-c.e. and n-c.e. equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e. and Π1 equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.