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 whichd 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 \(D\in\) d is just the minimum degree in which D is r.e. Then we study for a given d.r.e. degree d 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.
KeywordsMinimum Degree Minimum Element Striking Result Turing Degree
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