We show: for every noncomputable c.e. incomplete c-degree, there exists a nonspeedable c-degree incomparable with it; The c-degree of a hypersimple set includes an infinite collection of \-degrees linearly ordered under \ with order type of the integers and consisting entirely of hypersimple sets; there exist two c.e. sets having no c.e. least upper bound in the \-reducibility ordering; the c.e. \-degrees are not dense.

We show that the Q-degree of a hyperhypersimple set includes an infinite collection of Q1-degrees linearly ordered under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\leq_{Q_1}}$$\end{document} with order type of the integers and consisting entirely of hyperhypersimple sets. Also, we prove that the c.e. Q1-degrees are not an upper semilattice. The main result of this paper is that the Q1-degree of a hemimaximal set contains only one c.e. 1-degree. Analogous results are valid for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...) \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^0}$$\end{document}s1-degrees. (shrink)

We show that for every intermediate \ s-degree there exists an incomparable \ s-degree. As a consequence, for every intermediate \ Q-degree there exists an incomparable \ Q-degree. We also show how these results can be applied to provide proofs or new proofs of upper density results in local structures of s-degrees and Q-degrees.

We prove that if $A$, $B$ are noncomputable c.e. sets, $A<_{sQ_{1}}B$ and [($B$ is not simple and $A\oplus B\leq _{sQ_{1}}B$) or $B\equiv _{sQ_{1}}B\times \omega $], then there exist infinitely many pairwise $sQ_{1}$-incomparable c.e. sets $\{C_{i}\}_{i\in \omega }$ such that $A<_{sQ_{1}}C_{i}<_{sQ_{1}}B$, for all $i\in \omega $. We also show that there exist infinite collections of $sQ_{1}$-degrees $\{\boldsymbol {a_{i}}\}_{i\in \omega }$ and $\{\boldsymbol {b_{i}}\}_{i\in \omega }$ such that for every $i, j,$ (1) $\boldsymbol {a_{i}}<_{sQ_{1}}\boldsymbol {a_{i+1}}$, $\boldsymbol {b_{j+1}}<_{sQ_{1}}\boldsymbol {b_{j}}$ and $\boldsymbol {a_{i}}<_{sQ_{1}}\boldsymbol {b_{j}}$; (...) (2) every c.e set in $\boldsymbol {a_{i}}$ is a maximal set; and (3) every c.e. set in $\boldsymbol {b_{j}}$ is a non-maximal hyperhypersimple set. We prove that a noncomputable c.e. $sQ$-degree contains either only one or infinitely many c.e. $sQ_{1}$-degrees. The $sQ_{1}$-degrees of c.e. cylinders are dense upper semilattice. (shrink)

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{K}\not\le_{\rm ss} B}$$\end{document} (respectively, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{K}\not\le_{\overline{\rm s}} B}$$\end{document}): here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\le_{\overline{\rm (...) s}}}$$\end{document} is the finite-branch version of s-reducibility, ≤ss is the computably bounded version of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\le_{\overline{\rm s}}}$$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{K}}$$\end{document} is the complement of the halting set. Restriction to \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} sets provides a similar characterization of the \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} hyperhyperimmune sets in terms of s-reducibility. We also show that no \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \geq_{\overline{\rm s}}\overline{K}}$$\end{document} is hyperhyperimmune. As a consequence, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\deg_{\rm s}(\overline{K})}$$\end{document} is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed. (shrink)

We show that if M is an r‐maximal set, A is a major subset of M, B is an arbitrary set and, then. We prove that the c.e. ‐degrees are not dense. We also show that there exist infinite collections of ‐degrees and such that the following hold: (i) for every i, j,, and,(ii) each consists entirely of r‐maximal sets, and(iii) each consists entirely of non‐r‐maximal hyperhypersimple sets.