On the r.e. predecessors of d.r.e. degrees

Let d be a Turing degree containing differences of recursively enumerable sets (d.r.e.sets ) and R[d] be the class of less than d r.e.degrees in which d is relatively enumerable (r.e.). A.H.Lachlan proved that for any non-recursive d.r.e. d R[d] is not empty. We show that the r.e.degree defined by Lachlan for a d.r.e.set D ∈d is just the minimum degree in which D is r.e. Then we study for a given d.r.e.degree d the class R[d] and show that there exists a d.r.e. d such that R[d] has a minimum element >0. The most striking result of the paper is the existence of d.r.e.degrees for which R[d] consists of one element. Finally we prove that for some d.r.e. d R[d] can be the interval [a,b] for some r.e.degrees a,b, a<b<d .