Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-19T04:30:44.156Z Has data issue: false hasContentIssue false
  • English
  • Français

A high c.e. degree which is not the join of two minimal degrees

Published online by Cambridge University Press:  12 March 2014

Matthew B. Giorgi
Matthew B. Giorgi*
Affiliation:
14 Swan Hill Cottages, Aylesbury Road, Cuddington, Buckinghamshire, HP18 0BE, UK. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct a high c.e. degree which is not the join of two minimal degrees and so refute Posner's conjecture that every high c.e. degree is the join of two minimal degrees. Additionally, the proof shows that there is a high c.e. degree a such that for any splitting of a into degrees b and c one of these degrees bounds a 1-generic degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 4 , December 2010 , pp. 1339 - 1358
Copyright
Copyright © Association for Symbolic Logic 2010

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]Epstein, Richard L., Degrees of Unsolvabitity: Structure and Theory, Lecture Notes in Mathematics, vol. 759, Springer-Verlag, Berlin, Heidelberg, New York, 1979.CrossRefGoogle Scholar
[2]Jockusch, Carl G. Jr., Simple proofs of some theorems on high degrees of unsohability, Canadian Journal of Mathematics, vol. 29 (1977), pp. 1072–1080.CrossRefGoogle Scholar
[3]Jockusch, Carl G. Jr. and Posner, David B., Double jumps of minimal degrees, this Journal, vol. 43 (1978), pp. 715–724.Google Scholar
[4]Posner, David B., Minimal degrees and high degrees, an unpublished paper, 1976.Google Scholar
[5]Posner, David B., High degrees, Ph.D. thesis, University of California, Berkeley, 1977.Google Scholar
[6]Posner, David B., A survey of non-r.e. degrees < 0′, Recursion theory: its generalisation and applications (proceedings of Logic Colloquium '79, Leeds) (Drake, Frank R. and Wainer, Stanley S., editors), Cambridge University Press, Cambridge, 1980, pp. 52–109.CrossRefGoogle Scholar