Abstract
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 1∩A 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambos-Spies, K.: Contiguous r.e. degrees. In: Logic Colloquium '83. Springer Lecture notes1104, 1–37 (1984).
Ambos-Spies, K.: Cupping and noncapping in the r.e. weak truth table and Turing degrees. Archiv für Math. Logik.25, 109–126 (1985).
Ambos-Spies, K.: Anti-mitotic recursively enumerable sets. Z. Math. Logik Grund. Math.31, 461–477 (1985).
Ambos-Spies, K., Fejer, P.: Degree-theoretic splitting properties of r.e. sets, (to appear).
Ambos-Spies, K., Jockusch, C., Shore, R., Soare, R.: An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Trans. A. M. S.281, 109–128 (1984).
Downey, R.G.: The degrees of r.e. sets without the universal splitting property. Trans. A. M. S.291, 337–351 (1985).
Downey, R., Ingrassia, M., Stob, M., Welch, L.: unpublished manuscript.
Downey, R.G., Jockusch, C.G.:T-degrees, jump classes and strong reducibilities, (to appear). Trans. A. M. S.
Downey, R.G., Stob, M.: Structural interactions of the recursively enumerableT- andW-degrees. Annals Pure and Applied Logic31, 205–236 (1986).
Downey, R.G., Welch, L.V.: Splitting properties of r.e. sets and degrees. J. S. L.51, 88–109 (1986).
Fejer, P., Soare, R.I.: The plus cupping theorem for the recursively enumerable degrees. In: Logic Year 1979–1980, (Ed. Lerman, Schmerl and Soare), pp. 49–62. New York: Springer 1981.
Ladner, R., Sasso, L.: The weak truth table degrees of recursively enumerable sets. Ann. Math. Logic4, 429–448 (1975).
Lerman, M., Remmel, J.B.: The universal splitting property I. In: Logic Colloquium '80, (Ed. Van Dalen, Lascar & Smiley), pp. 181–209, New York: North Holland 1982.
Lerman, M., Remmel, J.B.: The universal splitting property II. J. S. L.49, 137–150 (1984).
Soare, R.I.: The infinite injury priority method. J. S. L.41, 513–530 (1976).
Soare, R.I.: Recursively Enumerable Sets and Degrees, Springer-Verlag (Omega Series), New York (1987).
Stob, M.:Wtt-degrees andT-degrees of r.e. sets. J. S. L.48, 921–930 (1983).
Author information
Authors and Affiliations
Additional information
Research partially supported by N.U.S. Grant RP 85/83 (Singapore).
Rights and permissions
About this article
Cite this article
Downey, R.G. Localization of a theorem of Ambos-Spies and the strong anti-splitting property. Arch math Logik 26, 127–136 (1987). https://doi.org/10.1007/BF02017497
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02017497