The Journal of Symbolic Logic

Research Article

The strong anticupping property for recursively enumerable degrees  1

S. B. Cooper

Department of Mathematics, University of Leeds, Leeds LS2 9JT, England

Following Soare [11] we say a recursively enumerable (r.e.) degree a has the anticupping (a.c.) property if there is a nonzero r.e. degree b < a such that for no r.e. c < a does a = bc.

Cooper [2] and Yates showed that 0′ has the a.c. property, while Harrington (see Miller [6]) proved that every high r.e. degree a has the a.c. property.

The recent paper by Ambos-Spies, Jockusch, Shore and Soare [1] describes a general theoretical framework for cupping and capping below 0′ which seems likely to be useful in a wider context.

Definition. (1) We say b is strongly noncuppable below a if 0 < b < a and, for each d < a, b ∪ d ≠ a.

(2) We say an r.e. a has the strong anticupping property if there is an r.e. b which is strongly noncuppable below a.

The main results on cupping in (≤ 0′) are due to Epstein, Posner and Robinson. For instance it is known (Posner and Robinson [8]) that the s.a.c. property fails for 0′.

We prove below that r.e. degrees with the s.a.c. property do exist, hence obtaining a nonzero r.e. degree a such that (≤ a) ≢e (≤h) for any high r.e. degree h. This result, obtained by means of an infinite injury construction in (≤ 0′), extends Theorem 2 of [3], proved using a finite injury construction in (≤ 0′).

Our main source of notation and terminology is [3].

(Received June 04 1984)

(Revised July 29 1985)

(Revised January 27 1986)

(Revised January 06 1988)

Footnotes

1   We are grateful to the referee for a number of helpful suggestions and corrections. We are also grateful for valuable help from C. G. Jockusch and R. I. Soare in the preparation of the paper.