Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-11T03:05:55.756Z Has data issue: false hasContentIssue false
  • English
  • Français

Decomposition and infima in the computably enumerable degrees

Published online by Cambridge University Press:  12 March 2014

Rodney G. Downey ,
Geoffrey L. Laforte  and
Richard A. Shore
Rodney G. Downey
Affiliation:
School of Mathematical and Computing Sciences, Victoria University, P.O. BOX 600, Wellington, New Zealand, E-mail: Rod.Downey@mcs.vuw.ac.nz
Geoffrey L. Laforte
Affiliation:
Department of Computer Science, University of West Florida, 11000 University Parkway, Pensacola, FL 32514, USA, E-mail: glaforte@uwf.edu
Richard A. Shore
Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853, USA, E-mail: shore@math.cornell.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a ∪ c) ∩ (b ∪ c), a ∪ c ∣ b ∪ c, and c < a ∪ b.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003 , pp. 551 - 579
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ambos-Spies, K., On pairs of recursively enumerable degrees. Transactions of the American Mathematical Society, vol. 283 (1984), pp. 507–531.CrossRefGoogle Scholar
[2]Downey, R., The 0′′′ priority method with special attention to density results. Recursion theory week: Proceedings, Oberwolfach, 1989 (Ambos-Spies, K.et al., editors). Springer, Berlin, 1990.Google Scholar
[3]Fejer, P., The density of the nonbranching degrees. Annals of Pure and Applied Logic, vol. 24 (1983), pp. 113–130.CrossRefGoogle Scholar
[4]Kleene, S. C. and Post, E. L., The upper semi-lattice of degrees of recursive unsolvability, Annals of Mathematics, Second Series, vol. 59 (1954), pp. 370–407.CrossRefGoogle Scholar
[5]Lachlan, A., Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, Third Series, vol. 16 (1966), pp. 537–569.CrossRefGoogle Scholar
[6]Lachlan, A., A recursively enumerable degree which will not split over all lesser ones, Annals of Mathematical Logic, vol. 9 (1975), pp. 307–365.CrossRefGoogle Scholar
[7]Lachlan, A., Decomposition of recursively enumerable degrees. Proceedings of the American Mathematical Society, vol. 79 (1980), pp. 629–634.CrossRefGoogle Scholar
[8]Sacks, G., On the degrees less than 0′, Annals of Mathematics, Second Series, vol. 77 (1963), pp. 211–231.CrossRefGoogle Scholar
[9]Sacks, G., The recursively enumerable degrees are dense. Annals of Mathematics, Second Series, vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[10]Shore, R. A., A noninversion theorem for the jump operator, Annals of Pure and Applied Logic, vol. 40 (1988), pp. 277–303.CrossRefGoogle Scholar
[11]Shore, R. A. and Slaman, T. A., Working below a low2 recursively enumerably degree, Archive for Mathematical Logic, vol. 29 (1990), pp. 201–211.CrossRefGoogle Scholar
[12]Slaman, T. A., The density of infima in the recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 155–179.CrossRefGoogle Scholar
[13]Soare, R., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[14]Spector, C., On degrees of recursive unsolvability, Annals of Mathematics, vol. 64 (1956), pp. 581–592.CrossRefGoogle Scholar
[15]Yates, C. E. M., A minimal pair of recursively enumerable degrees, this Journal, vol. 31 (1966), pp. 159–168.Google Scholar