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An almost-universal cupping degree

Published online by Cambridge University Press:  12 March 2014

Jiang Liu  and
Guohua Wu
Jiang Liu*
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
Guohua Wu
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore, E-mail: guohua@ntu.edu.sg
*
State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, P.R. China, E-mail: liuj@ios.ac.cn
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Abstract

Say that an incomplete d.r.e. degree has almost universal cupping property, if it cups all the r.e. degrees not below it to 0′. In this paper, we construct such a degree d, with all the r.e. degrees not cupping d to 0′ bounded by some r.e. degree strictly below d. The construction itself is an interesting 0″′ argument and this new structural property can be used to study final segments of various degree structures in the Ershov hierarchy.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1137 - 1152
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Arslanov, M. M., Structural properties of the degrees below 0′, Doklady Akademiya Nauk SSSR. New Series, vol. 283 (1985), pp. 270–273.Google Scholar
[2]Arslanov, M. M., On structural properties of the degrees of Ershov's hierarchy, Izvestiya Vysshikh Uchebnykh Zavedeniǐ Matematika, vol. 7 (1988), pp. 27–33.Google Scholar
[3]Cooper, S. B., On a theorem of C. E. M. Yates, handwritten notes, 1974.Google Scholar
[4]Cooper, S. B., Harrington, L., Lachlan, A. H., Lempp, S., and Soare, R. I., The d.r.e. degrees are not dense, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 125–151.CrossRefGoogle Scholar
[5]Cooper, S. B. and Yi, X., Isolated d.r.e. degrees, preprint series, no. 17, 25pp., 1995, University of Leeds, Department of Pure Mathematics.Google Scholar
[6]Li, A., Song, Y., and Wu, G., Universal cupping degrees, Theory and applications of models of computation (Cai, Jin yi, Cooper, S. Barry, and Li, Angsheng, editors), Lecture Notes in Computer Science, 3959, 2006, pp. 721–730.CrossRefGoogle Scholar
[7]Li, A. and Yi, X., Cupping the recursively enumerable degrees by d.r.e. degrees, Proceedings of the London Mathematical Society, vol. 78 (1999), pp. 1–21.CrossRefGoogle Scholar
[8]Odifreddi, P., Classical recursion theory, Studies in Logic and the Foundations of Mathematics 125, North-Holland, Amsterdam, 1989.Google Scholar
[9]Slaman, T. A. and Steel, J. R., Complementation in the Turing degrees, this Journal, vol. 54 (1989), pp. 160–176.Google Scholar
[10]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[11]Wu, G., Isolation and the jump operator, Mathematical Logic Quarterly, vol. 47 (2001), pp. 525–534.3.0.CO;2-7>CrossRefGoogle Scholar
[12]Wu, G., Isolation and lattice embeddigs, this Journal, vol. 67 (2002), pp. 1055–1064.Google Scholar
[13]Wu, G., Bi-isolation in the d.c.e. degrees, this Journal, vol. 69 (2004), pp. 409–420.Google Scholar
[14]Wu, G., Jump operator and Yates degrees, this Journal, vol. 71 (2006), pp. 252–264.Google Scholar
[15]Yates, C. E. M., Recursively enumerable degrees and the degrees less than 0(1), Sets, models and recursion theory, Proceedings of the summer school in mathematical logic and tenth logic colloquium, Leicester, 1965 (Crossley, J. N., editor), North-Holland, Amsterdam, 1967, pp. 264–271.Google Scholar