CEA Operators and the Ershov Hierarchy

Abstract

We examine the relationship between the CEA hierarchy and the Ershov hierarchy within $\Delta_2^0$ Turing degrees. We study the long-standing problem raised in [1] about the existence of a low computably enumerable (c.e.) degree $\bf a$ for which the class of all non-c.e. $CEA(\bf a)$ degrees does not contain 2-c.e. degrees. We solve the problem by proving a stronger result: there exists a noncomputable low c.e. degree $\textbf{a}$ such that any $CEA(\bf a)$$\omega$-c.e. degree is c.e. Also we discuss related questions and possible generalizations of this result.