Say that a d.c.e. degree d is isolated by a c.e. degree b, if bMathematics Subject Classification (2000): 03D25, 03D30, 03D35 RID=""ID="" Key words or phrases: Computably enumerable (...) set – d.c.e. degree – Isolation – High/low hierarchy RID=""ID="" Ishmukhametov's research is supported by RFBR grant 01-01-00733, and Wu's research is supported by the Marsden Fund of New Zealand. Wu would like to thank his supervisor, Prof. Rod Downey, for his many helpful suggestions and comments. (shrink)
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. (shrink)
In "Bounding minimal degrees by computably enumerable degrees" by A. Li and D. Yang, (this Journal, ), the authors prove that there exist non-computable computably enumerable degrees c > a > 0 such that any minimal degree m being below c is also below a. We analyze the proof of their result and show that the proof contains a mistake. Instead we give a proof for the opposite result.
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  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. (shrink)