Department of Mathematics, University of Leeds, Leeds LS2 9JT, England
Following Soare  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 = b ∪ c.
Cooper  and Yates showed that 0′ has the a.c. property, while Harrington (see Miller ) proved that every high r.e. degree a has the a.c. property.
The recent paper by Ambos-Spies, Jockusch, Shore and Soare  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 ) 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 , proved using a finite injury construction in (≤ 0′).
Our main source of notation and terminology is .
(Received June 04 1984)
(Revised July 29 1985)
(Revised January 27 1986)
(Revised January 06 1988)
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.