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 connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We say that a class of computably enumerable sets bounded iff there is an m-incomplete computably enumerable set A such that every set in is m-reducible to A. For example, we show that the class of effectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded (...) iff B is low2. For r = bwtt,tt,wtt and T, there is a bounded class intersecting every computably enumerable r-degree; for r = c, d and p, no such class exists. (shrink)

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 solve a longstanding question of Soare by showing that if ${\mathbf d}$ is a non-low $_2$ computably enumerable degree then ${\mathbf d}$ contains a c.e. set with no r-maximal c.e. superset.

Downey and Lempp 1215–1240) have shown that the contiguous computably enumerable degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truth-table degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5-element lattice N5 into the c.e. degrees which maps the top to the degree a. In particular, this shows that local nondistributivity coincides with (...) local nonmodularity in the computably enumerable degrees. (shrink)