Localization of a theorem of Ambos-Spies and the strong anti-splitting property

LetA be an r.e. nonrecursive set. We sayA has thestrong antisplitting property if there exists an r.e. setB with 0< _T B< _T A such that ifA ₁ A ₂=A andA ₁A ₂=0 thenA _{1 T} B impliesA _{1 T} 0 andB _T A ₁ impliesA _{1 T} A. It is shown that below any high r.e. degree there exists an r.e. set with the strong antisplitting property. The main ingredient of the proof is a localization of Ambos-Spies' result that the cup or cap theorem fails forW-degrees.Research partially supported by N.U.S. Grant RP 85/83 (Singapore).