On distributivity of the upper semilattices of the degrees below \(0'\) (Q1902712)

Let \(D^r_1\) and \(D^r_2\) denote the upper semilattices of the recursively enumerable (r.e.) \(r\)-degrees and of the \(r\)-degrees of differences of r.e. sets, respectively, where \(r\) is one of the reducibilities which have been studied in recursive function theory. Numerous differences in the structure of these upper semilattices in the case of Turing reducibility give rise to the interest for studying the structure of \(D^r_2\) for other reducibilities \(r\). We prove here that the structures \(D^r_2\) are not distributive for any reducibility \(r\) intermediate between bd and T.