CEA-Operators and the Ershov Hierarchy. I

We consider the relationship between the CEA-hierarchy and the Ershov hierarchy in \({\Delta }_{2}^{0}\) Turing degrees. A degree c is called CEA(a) if c is computably enumerable in a, and a ≤ c. Soare and Stob [Stud. Logic Found. Math., 107, 299-324 (1982)] proved that for a noncomputable low c.e. degree a there exists a CEA(a) degree that is not c.e. Later, Arslanov, Lempp, and Shore [Ann. Pure Appl. Logic, 78, Nos. 1-3, 29-56 (1996)] formulated the problem of describing pairs of degrees a < e such that there exists a CEA(a) 2-c.e. degree d ≤ e which is not c.e. Since then the question has remained open as to whether a CEA(a) degree in the sense of Soare and Stob can be made 2-c.e. Here we answer this question in the negative, solving it in a stronger formulation: there exists a noncomputable low c.e. degree a such that any CEA(a) ω-c.e. degree is c.e. Also possible generalizations of the result obtained are discussed, as well as various issues associated with the problem mentioned.