We give an algorithm for deciding whether an embedding of a finite partial order [Formula: see text] into the enumeration degrees of the [Formula: see text]-sets can always be extended to an embedding of a finite partial order [Formula: see text].

We show that every splitting of ${0}_{\mathrm{e}}^{\prime }$ in the local structure of the enumeration degrees, $$\mathcal{G}_{e} , contains at least one low-cuppable member. We apply this new structural property to show that the classes of all $\mathcal{K}$ -pairs in $\mathcal{G}_{e}$ , all downwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees and all upwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees are first order definable in $\mathcal{G}_{e}$.

We show that in the language of {≤}, the Π₃-fragment of the first order theory of the $\Sigma _{2}^{0}$-enumeration degrees is undecidable. We then extend this result to show that the Π₃-theory of any substructure of the enumeration degrees which contains the $\Delta _{2}^{0}$-degrees is undecidable.

We give an alternative and more informative proof that every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree is the meet of two incomparable ${\Sigma^{0}_{2}}$ -degrees, which allows us to show the stronger result that for every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree a, there exist enumeration degrees x 1 and x 2 such that a, x 1, x 2 are incomparable, and for all b ≤ a, b = (b ∨ x 1 ) ∧ (b ∨ x 2 ).

We investigate strong versions of enumeration reducibility, the most important one being s-reducibility. We prove that every countable distributive lattice is embeddable into the local structure $L(\mathfrak D_s)$ of the s-degrees. However, $L(\mathfrak D_s)$ is not distributive. We show that on $\Delta^{0}_{2}$ sets s-reducibility coincides with its finite branch version; the same holds of e-reducibility. We prove some density results for $L(\mathfrak D_s)$ . In particular $L(\mathfrak D_s)$ is upwards dense. Among the results about reducibilities that are stronger than s-reducibility, (...) we show that the structure of the $\Delta^{0}_{2}$ bs-degrees is dense. Many of these results on s-reducibility yield interesting corollaries for Q-reducibility as well. (shrink)

We prove that for every ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degree b there exists a noncuppable ${\mathrm{\Sigma }}_{2}^{0}$ degree a > 0 e such that b′ ≤ e a′ and a″ ≤ e b″. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding l: D T → D e , that there exist ${\mathrm{\Sigma }}_{2}^{0}$ noncuppable enumeration degrees at every possible—i.e., above low₁—level of the high/low (...) jump hierarchy in the context of D e. (shrink)

We show that there is a first order sentence φ(x; a, b, l) such that for every computable partial order.

For any enumeration degree let be the set of s-degrees contained in . We answer an open question of Watson by showing that if is a nontrivial -enumeration degree, then has no least element. We also show that every countable partial order embeds into . Finally, we construct -sets A and B such that B≤eA but for every X≡eB, XsA.

The jump operator on the ω-enumeration degrees was introduced in [I.N. Soskov, The ω-enumeration degrees, J. Logic Computat. 17 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of and of the enumeration degrees are isomorphic. In the second part of the paper we (...) study the jumps of the ω-enumeration degrees below . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω-enumeration degrees below which are high n for some n and of the class L of the ω-enumeration degrees below which are low n for some n. (shrink)

This paper continues the investigation into the relationship between good approximations and jump inversion initiated by Griffith. Firstly it is shown that there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^{0}_{2}}$$\end{document} set A whose enumeration degree a is bad—i.e. such that no set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \in a}$$\end{document} is good approximable—and whose complement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{A}}$$\end{document} has lowest possible jump, in other words (...) is low2. This also ensures that the degrees y ≤ a only contain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{3}}$$\end{document} sets and thus yields a tight lower bound for the complexity of both a set of bad enumeration degree, and of its complement, in terms of the high/low jump hierarchy. Extending the author’s previous characterisation of the double jump of good approximable sets, the triple jump of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{0}_{2}}$$\end{document} set A is characterised in terms of the index set of coinfinite sets enumeration reducible to A. The paper concludes by using Griffith’s jump interpolation technique to show that there exists a high quasiminimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{2}}$$\end{document} enumeration degree. (shrink)

Defining a class of sets to be uniform Δ02 if it is derived from a binary {0,1}{0,1}-valued function f≤TKf≤TK, we show that, for any C⊆DeC⊆De induced by such a class, there exists a high Δ02 degree c which is incomparable with every degree b ϵ Ce \ {0e, 0'e}. We show how this result can be applied to quite general subclasses of the Ershov Hierarchy and we also prove, as a direct corollary, that every nonzero low degree caps with both (...) a high and a nonzero low Δ02 degree. (shrink)

We complete a study of the splitting/non-splitting properties of the enumeration degrees below by proving an analog of Harrington’s non-splitting theorem for the enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.

We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps are enumeration degree jumps.) (...) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees. (shrink)

We show that every nonzero $\Sigma _{2}^{0}$ enumeration degree bounds a nonsplitting nonzero enumeration degree.

We prove that a sequence of sets containing representatives of cupping partners for every nonzero ${\Delta^0_2}$ enumeration degree cannot have a ${\Delta^0_2}$ enumeration. We also prove that no subclass of the ${\Sigma^0_2}$ enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to ${\mathbf{0}_e'}$ by a single incomplete ${\Sigma^0_2}$ enumeration degree.

We investigate and extend the notion of a good approximation with respect to the enumeration ${({\mathcal D}_{\rm e})}$ and singleton ${({\mathcal D}_{\rm s})}$ degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index sets, in the context of enumeration reducibility. We study partial order embeddings ${\iota_s}$ and ${\hat{\iota}_s}$ of, respectively, ${{\mathcal D}_{\rm e}}$ and ${{\mathcal D}_{\rm T}}$ (the Turing degrees) into (...) ${{\mathcal D}_{\rm s}}$ , and we show that the image of ${{\mathcal D}_{\rm T}}$ under ${\hat{\iota}_s}$ is precisely the class of retraceable singleton degrees. We define the notion of a good enumeration, or singleton, degree to be the property of containing the set of good stages of some good approximation, and we show that ${\iota_s}$ preserves the latter, as also other naturally arising properties such as that of totality or of being ${\Gamma^0_n}$ , for ${\Gamma \in \{\Sigma,\Pi,\Delta\}}$ and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good ${\Sigma^0_2}$ singleton degrees are hyperimmune. Finally we show that, for singleton degrees a s < b s such that b s is good, any countable partial order can be embedded in the interval (a s, b s). (shrink)