On the structure of computable reducibility on equivalence relations of natural numbers

Abstract

We examine the degree structure $\mathbf{ER}$ of equivalence relations on $\omega$ under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not have a join but that some incomparable degrees do, and we characterize the degrees which have a join with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in $\mathbf{ER}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf{d}\oplus \mathbf{I}{\mathbf{d}}_1$ needn't be a strong minimal cover of a self-full degree $\mathbf{d}$. Finally, we show that the theory of the degree structure $\mathbf{ER}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second order arithmetic.

30 pages, accepted for publication in The Journal of Symbolic Logic

References in corpus (1)

• Computably Enumerable Equivalence Relations

Cited by in corpus (1)

• Computable reducibility of equivalence relations and an effective jump operator