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

Abstract

We examine the degree structure ER of equivalence relations on ω 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 ER. We show that every equivalence relation has continuum many self-full strong minimal covers, and that d ' Id1 needn't be a strong minimal cover of a self-full degree d. Finally, we show that the theory of the degree structure ER as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second order arithmetic.2000 Mathematics Subject Classification. 03D55.

Punctual equivalence relations and their (punctual) complexity

“…We prove now that the distributive lattice of punctual degrees is dense. This contrasts with the case of Ceers and ER, where each degree has a minimal cover (see [6,3] for details). However, density is a phenomenon that often shows up when focusing on the subrecursive world.…”

“…Less is known about larger structures of c-degrees; but recent studies considered the ∆ 0 2 case [30,9] and the global structure ER of all c-degrees [3]. Yet, despite its classificatory power, computable reducibility has an obvious shortcoming: it is simply too coarse for measuring the relative complexity of computable equivalence relations.…”