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)

A result of Soare and Stob asserts that for any non-recursive r.e. setC, there exists a r.e.[C] setA such thatA⊕C is not of r.e. degree. A setY is called [of]m-REA (m-REA[C] [degree] iff it is [Turing equivalent to] the result of applyingm-many iterated ‘hops’ to the empty set (toC), where a hop is any function of the formX→X ⊕W e X . The cited result is the special casem=0,n=1 of our Theorem. Form=0,1, and any (m+1)-REA setC, ifC is not ofm-REA (...) degree, then for alln there exists an-r.e.[C] setA such thatA ⊕C is not of (m+n)-REA degree. We conjecture that this holds also form≥2. (shrink)

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

In the present paper we prove that the isolated differences of r. e. degrees are dense in the r. e. degrees.

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.