On the Structure of Computable Reducibility on Equivalence Relations of Natural Numbers

Abstract

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

Investigating the Computable Friedman–stanley Jump

“…In this paper, we answer several of these questions. By doing so, we advance understanding of both the computable Friedman-Stanley jump and the global hierarchy of countable equivalence relations explored in [3]. We will prove, in particular, that the computable Friedman-Stanley tower is a cofinal family of HYP equivalence relations (Corollary 6.13).…”

Computable Reducibility of Equivalence Relations and an Effective Jump Operator

“…The following characterization then follows from the fact that a set is hyperarithmetic if and only if it is Δ 1 1 , together with the Kleene Separation Theorem and the hyperarithmetic codes used in its proof (see, e.g., [17, Chapter II] and [15, Theorem 27.1]).…”

“…Here we can use any reasonable coding of computable trees by natural numbers. Then ∼ =T is a Σ 1 1 equivalence relation which is not hyperarithmetic. In [10, Theorem 2] it was shown that ∼ =T is Σ 1 1 complete for computable reducibility, that is, ∼ =T is Σ We note that although every hyperarithmetic set is many-one reducible to Id +a for some a ∈ O, we do not know whether every hyperarithmetic equivalence relation E satisfies E ≤ Id +a for some a ∈ O.…”

“…1 1 and for every Σ1 1 equivalence relationE, E ≤ ∼ =T . We can see that ∼ =T is a jump fixed point, i.e., ∼ = + T is computably bireducible with ∼ =T .…”

Investigating the computable Friedman-Stanley jump

“…Note that Id `is equivalent is to " ce , where x " ce y if and only if W x " W y . Following [AS19,ABSM], we say that:…”