Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-22T20:45:24.112Z Has data issue: false hasContentIssue false
  • English
  • Français

Bi-isolation in the d.c.e. degrees

Published online by Cambridge University Press:  12 March 2014

Guohua Wu
Guohua Wu*
Affiliation:
School of Mathematical and Computing Sciences, Victoria University of Wellington, P. O. BOX 600, Wellington, New Zealand, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract.

In this paper, we study the bi-isolation phenomena in the d.c.e. degrees and prove that there are c.e. degrees c1 < c2 and a d.c.e. degree d ∈ (c1, c2) such that (c1, d) and (d, c2) contain no c.e. degrees. Thus, the c.e. degrees between c1 and c2 are all incomparable with d. We also show that there are d.c.e. degrees d1 < d2 such that (d1, d2) contains a unique c.e. degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 2 , June 2004 , pp. 409 - 420
Copyright
Copyright © Association for Symbolic Logic 2004

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]Arslanov, M. M., Structural properties of the degrees below 0′, Doklady Akademii Nauk UzSSR, vol. 283 (1985), pp. 270–273.Google Scholar
[2]Arslanov, M. M., private communications, 01 2002.Google Scholar
[3]Arslanov, M. M., Lempp, S., and Shore, R. A., On isolating r.e. and isolated d.r.e. degrees, Computability, enumerability, unsolvability (Cooper, S. B., Slaman, T. A., and Wainer, S. S., editors), 1996, pp. 61–80.CrossRefGoogle Scholar
[4]Cooper, S. B., A splitting theorem of the n-r.e. degrees, Proceedings of the American Mathematical Society, vol. 115, pp. 461–471.Google Scholar
[5]Cooper, S. B., Degrees of unsolvability, Ph. D. thesis, Leicester University, 1971.Google Scholar
[6]Cooper, S. B., The density of the low2 n-r.e. degrees, Archive for Mathematical Logic, vol. 30 (1991), pp. 19–24.CrossRefGoogle Scholar
[7]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
[8]Cooper, S. B., Lempp, S., and Watson, P., Weak density and cupping in the d-r.e. degrees, Israel Journal of Mathematics, vol. 67 (1989), pp. 137–152.CrossRefGoogle Scholar
[9]Cooper, S. B. and Yi, X., Isolated d.r.e. degrees, preprint series 17, University of Leeds, Department of Pure Mathematics, 1995, 25 pp.Google Scholar
[10]Ding, D. and Qian, L., Isolated d.r.e. degrees are dense in r.e. degree structure, Archive for Mathematical Logic, vol. 36 (1996), pp. 1–10.CrossRefGoogle Scholar
[11]Downey, R. G., D.r.e. degrees and the nondiamond theorem, Bulletin of the London Mathematical Society, vol. 21 (1989), pp. 43–50.CrossRefGoogle Scholar
[12]Efremov, A., Upper isolated d.c.e. degrees, I, Izvestiya Vysshikh Uchebnykh Zavedeniĭ, Matematika, vol. 42 (1998), no. 2, pp. 20–28, in Russian.Google Scholar
[13]Efremov, A., Upper isolated d.c.e. degrees, II, Izvestiya Vysshikh Uchebnykh Zavedeniĭ, Matematika, vol. 42 (1998), no. 7, pp. 18–25, in Russian.Google Scholar
[14]ErsHoV, Yu. L., On a hierarchy of sets I, Algebra i Logika, vol. 7 (1968), pp. 47–73.Google Scholar
[15]Harrington, L. and Soare, R. I., Games in recursion theory and continuity properties of capping degrees, Set theory and the continuum (Judah, H., Just, W., and Woodin, W. H., editors), 1992, pp. 39–62.CrossRefGoogle Scholar
[16]Ishmukhametov, S., D.r.e. sets, their degrees and index sets, Thesis, Novosibirsk, Russia, 1986.Google Scholar
[17]Ishmukhametov, S. and Wu, G., Isolation and the high/low hierarchy, Archive for Mathematical Logic, vol. 41 (2002), pp. 259–266.CrossRefGoogle Scholar
[18]LaForte, G., The isolated d.r.e. degrees are dense in the r.e. degrees, Mathematical Logic Quarterly, vol. 42 (1996), pp. 83–103.CrossRefGoogle Scholar
[19]Lempp, S., private communications, 04 2001.Google Scholar
[20]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
[21]Sacks, G. E., On the degrees less than 0′, Annals of Mathematics, vol. 77 (1963), pp. 211–231.CrossRefGoogle Scholar
[22]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[23]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[24]Wu, G., Isolation and the jump operator, Mathematical Logic Quarterly, vol. 47 (2001), pp. 525–534.3.0.CO;2-7>CrossRefGoogle Scholar
[25]Wu, G., Isolation and lattice embeddings, this Journal, vol. 67 (2002), pp. 1055–1064.Google Scholar
[26]Wu, G., Nonisolated degrees and the jump operator, Annals of Pure and Applied Logic, vol. 117 (2002), pp. 209–221.CrossRefGoogle Scholar