Bounding computably enumerable degrees in the Ershov hierarchy

Abstract

Lachlan observed that any nonzero d.c.e. degree bounds a nonzero c.e. degree. In this paper, we study the c.e. predecessors of d.c.e. degrees, and prove that given a nonzero d.c.e. degree a, there is a c.e. degree b below a and a high d.c.e. degree d > b such that b bounds all the c.e. degrees below d. This result gives a unified approach to some seemingly unrelated results. In particular, it has the following two known theorems as corollaries: (1) there is a low c.e. degree isolating a high d.c.e. degree [S. Ishmukhametov, G. Wu, Isolation and the high/low hierarchy, Arch. Math. Logic 41 (2002) 259-266]; (2) there is a high d.c.e. degree bounding no minimal pairs [C.T. Chong, A. Li, Y. Yang, The existence of high nonbounding degrees in the difference hierarchy, Ann. Pure Appl. Logic 138 (2006) 31-51]. © 2005 Elsevier B.V. All rights reserved.