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

TL;DR

This paper explores the structure of computable reducibility among equivalence relations on natural numbers, analyzing joins, definability, and the logical complexity of the degree structure.

Contribution

It characterizes when degrees have joins, definability of classes, and the logical theories of the degree structure and its substructures.

Findings

Incomparable degrees may or may not have joins.

Natural classes of degrees are definable within the structure.

Theories of the degree structure and substructures are computably isomorphic to second order arithmetic.

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 $\mathbf{d}\oplus \mathbf{Id_1}$ 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.