Abstract

Let d be a Turing degree, R[d] and Q[d] denote respectively classes of recursively enumerable (r.e.) and all degrees in which d is relatively enumerable. We proved in Ishmukhametov [1999] that there is a degree d containing differences of r.e.sets (briefly, d.r.e.degree) such that R[d] possess a least elementm $>$ 0. Now we show the existence of a d.r.e. d such that R[d] has no a least element. We prove also that for any REA-degree d below 0 $'$ the class Q[d] cannot have a least element and more generally is not bounded below by a non-zero degree, except in the trivial cases