Intervals containing exactly one c.e. degree

Abstract

Cooper proved in [S.B. Cooper, Strong minimal covers for recursively enumerable degrees, Math. Logic Quart. 42 (1996) 191–196] the existence of a c.e. degree a with a strong minimal cover d . So a is the greastest c.e. degree below d . Cooper and Yi pointed out in [S.B. Cooper, X. Yi, Isolated d.r.e. degrees, University of Leeds, Dept. of Pure Math., 1995. Preprint] that this strongly minimal cover d cannot be d.c.e., and meanwhile, they proposed the notion of isolated degrees: a d.c.e. degree d is isolated by a c.e. degree a if a is the greatest c.e. degree below d , and we also say that a isolates d . In [G. Wu, Bi-isolation in the d.c.e. degrees, J. Symbolic Logic 69 (2004) 409–420], Wu extended Cooper–Yi’s notion and proved that there are intervals of d.c.e. degrees ( d 1 , d 2 ) containing exactly one c.e. degree a . Following Cooper and Yi’s notion, a is called a bi-isolating degree. The bi-isolating degrees are dense in the high c.e. degrees. Arslanov asked whether the bi-isolating degrees occur in every jump class. In this paper, we prove that there are low bi-isolating degrees, providing a partial solution to Arslanov’s question.