Abstract:
In this thesis, we are mainly concerned with the structural properties of the d.c.e. degrees and the distribution of the simple reals among the c.e. degrees.
In chapters 2 and 3, we study the relationship between the isolation phenomenon and the jump operator. We prove in chapter 2 that there is a high d.c.e. degree d isolated by a low2 degree a. We improve this result in chapter 3 by showing that the isolating degree a can be low.
Chapters 4 and 5 are devoted to the study of the pseudo-isolation in the d.c.e. degrees. We prove that pseudo-isolated d.c.e. degrees are dense in the c.e. degrees, and that there is a high d.c.e. degree pseudo-isolated by a low d.c.e. degree.
In chapter 6, we prove that there are two d.c.e. degrees between which there is exactly one c.e. degree and that there are two c.e. degree a1 < a2 and a d.c.e. degree d ∈ (a1, a2) such that the intervals (a1, d) and (d, a2) contains no c.e. degrees.
In chapter 7, we construct an isolation pair (a, d) and a c.e. degree c such that c ∪ d = 0', c ∩ a = 0. This result gives an alternative proof of Downey's diamond embedding into the d.c.e. degrees.
Chapter 8 is a contribution to the study of the distribution of the simple reals in the c.e. degrees. We show that there is a noncomputable c.e. degree bounding no noncomputable the simple reals. Thus, simple reals are not dense in the structure of the computably enumerable degrees.
