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.
Wu is partially supported by a research grant No. RG58/06 (M52110023.710079) from NTU.
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References
Cholak, P., Groszek, M., Slaman, T.: An almost deep degree. J. Symbolic Logic 66, 881–901 (2001)
Jockusch Jr., C.G., Li, A., Yang, Y.: A join theorem for the computably enumerable degrees. Trans. Amer. Math. Soc. 356, 2557–2568 (2004)
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Liu, J., Wu, G. (2008). Joining to High Degrees. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds) Logic and Theory of Algorithms. CiE 2008. Lecture Notes in Computer Science, vol 5028. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69407-6_40
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DOI: https://doi.org/10.1007/978-3-540-69407-6_40
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