Infima in the Recursively Enumerable

x0. Introduction and Notation. Weak truth table reducibility (w-reducibility) was rst introduced by Friedberg and Rogers [FR59]. Intuitively, we say that a set A is w-reducible to a set B (written A w B) if there is a Turing reduction from A to B and a recursive function f such that, for any x, the value f(x) bounds the greatest number whose membership or nonmembership in B is used to determine A(x). Since w-reducibility is a stronger reducibility than Turing reducibility, each Turing degree can be partitioned into the w-degrees of its sets. Ladner and Sasso [LS75] showed that there exists a nonzero contiguous degree, i.e., an r.e. Turing degree which contains a single r.e. w-degree. The existence of such contiguous degrees, as well as the strongly contiguous degrees introduced by Downey [Do87], has been used to establish numerous existence results (cf. [LS75], [St83], [Am84a]) in the r.e. Turing degrees by establishing the corresponding results in the r.e. w-degrees. Our results here deal with in ma in the r.e. w-degrees, thus continuing the investigations of Cohen [Co75], Ambos-Spies [Am85], and Fischer [Fi86]. Cohen's result that every incomplete r.e. w-degree is w-branching and Fischer's result that some initial segments of the r.e. w-degrees are lattices indicate that in ma are more common in the r.e. w-degrees than in the r.e. Turing degrees. We reinforce this notion.