Abstract
In this paper, we prove that for any nonzero cappable degree c, there is a d.c.e. degree d and a c.e. degree b < d such that c cups d to 0′, caps b to 0 and for any c.e. degree w, either w ≤ b or \({\bf w}\lor{\bf d}={\bf 0}\prime\). This result has several well-known theorems as direct corollaries, including Arslanov’s cupping theorem, Downey’s diamond theorem, Downey-Li-Wu’s complementation theorem, and Li-Yi’s cupping theorem, etc.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ambos-Spies, K., Jockusch Jr., C.G., Shore, R.A., Soare, R.I.: An algebraic decomposition of the recursively enumerable degrees and the coincidence of several classes with the promptly simple degrees. Trans. Amer. Math. Soc. 128, 109–128, 537-569 (1984)
Arslanov, M.M.: Structural properties of the degrees below 0′. Dokl. Akad, Nauk SSSR(N. S.) 283, 270–273 (1985)
Cooper, S.B., Harrington, L., Lachlan, A.H., Lempp, S., Soare, R.I.: The d.r.e. degrees are not dense. Ann. Pure Appl. Logic 55, 125–151 (1991)
Downey, R.: D.r.e. degrees and the nondiamond theorem. Bull. London Math. Soc. 21, 43–50 (1989)
Downey, R., Li, A., Wu, G.: Complementing cappable degrees in the difference hierachy. Annals of pure and applied logic 125, 101–118 (2004)
Li, A., Song, Y., Wu, G.: Universal cupping degrees. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 721–730. Springer, Heidelberg (2006)
Li, A., Yi, X.: Cupping the recursively enumerable degrees by d.r.e. degrees. Proc. London Math. Soc. 79, 1–21 (1999)
Liu, J., Wu, G.: Degrees with almost universal cupping property. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds.) CiE 2010. LNCS, vol. 6158, pp. 266–275. Springer, Heidelberg (2010)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)
Wu, G.: Isolation and lattice embeddings. Journal of Symbolic Logic 67, 1055–1064 (2002)
Yates, C.E.M.: A minimal pair of recursively enumerable degrees. J. Symbolic Logic 31, 159–168 (1966)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fang, C., Liu, J., Wu, G. (2011). Cupping and Diamond Embeddings: A Unifying Approach. In: Löwe, B., Normann, D., Soskov, I., Soskova, A. (eds) Models of Computation in Context. CiE 2011. Lecture Notes in Computer Science, vol 6735. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21875-0_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-21875-0_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21874-3
Online ISBN: 978-3-642-21875-0
eBook Packages: Computer ScienceComputer Science (R0)