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Joining to High Degrees

  • Jiang Liu
  • Guohua Wu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5028)

Abstract

Cholak, Groszek and Slaman proved in [1] that there is a nonzero computably enumerable (c.e.) degree cupping every low c.e. degree to a low c.e. degree. In the same paper, they pointed out that every nonzero c.e. degree can cup a low2 c.e. degree to a nonlow2 degree. In [2], Jockusch, Li and Yang improved the latter result by showing that every nonzero c.e. degree c is cuppable to a high c.e. degree by a low2 c.e. degree b. It is natural to ask in which subclass of low2 c.e. degrees b in [2] can be located. Wu proved [6] that b can be cappable. We prove in this paper that b in Jockusch, Li and Yang’s result can be noncuppable, improving both Jockusch, Li and Yang, and Wu’s results.

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References

  1. 1.
    Cholak, P., Groszek, M., Slaman, T.: An almost deep degree. J. Symbolic Logic 66, 881–901 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
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    Jockusch Jr., C.G., Li, A., Yang, Y.: A join theorem for the computably enumerable degrees. Trans. Amer. Math. Soc. 356, 2557–2568 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
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    Liu, J., Wu, G.: Joining to high degrees via noncuppables (in preparation)Google Scholar
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    Miller, D.: High recursively enumerable degrees and the anti-cupping property. In: Lerman, Schmerl, Soare (eds.) Logic Year 1979-80: University of Connecticut. Lecture Notes in Mathematics, vol. 859, pp. 230–245 (1981)Google Scholar
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    Wu, G.: Quasi-complements of the cappable degrees. Math. Log. Quart. 50, 189–201 (2004)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jiang Liu
    • 1
  • Guohua Wu
    • 1
  1. 1.School of Physical and Mathematical SciencesNanyang Technological UniversitySingapore

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