We consider the relationship of the lattice-theoretic properties and the jump-theoretic properties satisfied by a recursively enumerable Turing degree. The existence is shown of a high2 r.e. degree which does not bound what we call the base of any Slaman triple.

We study the jump hierarchy of d.c.e. Turing degrees and show that there exists a high d.c.e. degree d which does not bound any minimal pair of d.c.e. degrees.

We prove that every non-computable incomplete computably enumerable degree is locally non-cappable, and use this result to show that there is no maximal non-bounding computably enumerable degree.

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

We prove that for any computably enumerable degree c, if it is cappable in the computably enumerable degrees, then there is a d.c.e. degree d such that c d = 0′ and c ∩ d = 0. Consequently, a computably enumerable degree is cappable if and only if it can be complemented by a nonzero d.c.e. degree. This gives a new characterization of the cappable degrees.

Lachlan observed that any nonzero d.c.e. degree bounds a nonzero c.e. degree. In this paper, we study the c.e. predecessors of d.c.e. degrees, and prove that given a nonzero d.c.e. degree , there is a c.e. degree below and a high d.c.e. degree such that bounds all the c.e. degrees below . This result gives a unified approach to some seemingly unrelated results. In particular, it has the following two known theorems as corollaries: there is a low c.e. degree isolating (...) a high d.c.e. degree [S. Ishmukhametov, G. Wu, Isolation and the high/low hierarchy, Arch. Math. Logic 41 259–266]; there is a high d.c.e. degree bounding no minimal pairs [C.T. Chong, A. Li, Y. Yang, The existence of high nonbounding degrees in the difference hierarchy, Ann. Pure Appl. Logic 138 31–51]. (shrink)

The Major Sub-degree Problem of A. H. Lachlan (first posed in 1967) has become a long-standing open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, ${{\bf 0' = b \lor x}}$ if and only if (...) ${{\bf 0' = a \lor x}}$ . In this paper, we show that every c.e. degree b ≠ 0 or 0′ has a major sub-degree, answering Lachlan’s question affirmatively. (shrink)

We prove that the existential theory of the Turing degrees, in the language with Turing reduction, 0, and unary relations for the classes in the generalized high/low hierarchy, is decidable. We also show that every finite poset labeled with elements of (where is the partition of induced by the generalized high/low hierarchy) can be embedded in preserving the labels. Note that no condition is imposed on the labels.

In this article we establish the existence of a number of new orbits in the automorphism group of the computably enumerable sets. The degree theoretical aspects of these orbits also are examined.

We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w,v ≤ a, if a ≤ w or; v or; b then either a ≤ w or; b or a ≤ v or; b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy. We prove that there (...) is a non-Low2 c.e. degree which bounds no Lachlan nonsplitting base. (shrink)

In this paper, we improve a result of Seetapun and prove that above any nonzero, incomplete recursively enumerable degree a, there is a high2 r.e. degree c>ac>a witnessing that a is locally noncappable . Theorem 1.1 provides a scheme of obtaining high2 nonboundings , as all known high2 nonboundings, such as high2 degrees bounding no minimal pairs, high2 plus-cuppings, etc.

In this paper, we first prove that there exist computably enumerable (c.e.) degrees a and b such that ${\bf a\not\leq b}$ , and for any c.e. degree u, if ${\bf u\leq a}$ and u is cappable, then ${\bf u\leq b}$ , so refuting a conjecture of Lempp (in Slaman [1996]); secondly, we prove that: (A. Li and D. Wang) there is no uniform construction to build nonzero cappable degree below a nonzero c.e. degree, that is, there is no computable function (...) $f$ such that for all $e\in\omega,$ (i) $W_{f(e)}\leq_{\rm T}W_e$ , (ii) $W_{f(e)}$ has a cappable degree, and (iii) $W_{f(e)}\not\leq_{\rm T}\emptyset$ unless $W_e\leq_{\rm T}\emptyset.$. (shrink)

It will be shown that there exist computably enumerable degrees a and b such that a $>$ b, and for any computably enumerable degree u, if u $\leq$ a and u is cappable, then u $<$ b.