We use some simple facts about the wtt-degrees of r.e. sets together with a construction to answer some questions concerning the join and meet operators in the r.e. degrees. The construction is that of an r.e. Turing degree a with just one wtt-degree in a such that a is the join of a minimal pair of r.e. degrees. We hope to illustrate the usefulness of studying the stronger reducibility orderings of r.e. sets for providing information about Turing reducibility.

A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...) the structure of , the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory. (shrink)

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)