Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Joining up to the generalized high degrees

Author(s): Philip Ellison; Andrew E. M. Lewis
Journal: Proc. Amer. Math. Soc. 138 (2010), 2949-2960.
MSC (2000): Primary 03D28; Secondary 03D10
Posted: March 29, 2010
MathSciNet review: 2644906
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We show that every generalized high Turing degree is the join of two minimal degrees, thereby settling a conjecture of Posner's from the 70s.

References:

[BC]: S.B. Cooper, Degrees of unsolvability complementary between recursively enumerable degrees, Annals of Mathematical Logic, 4 (1), 31-74 (1972). MR 0294126 (45:3199)
[BC2]: S.B. Cooper, Minimal degrees and the jump operator, Journal of Symbolic Logic, 38 (2) (1973), 249-271. MR 0347572 (50:75)
[N4]: R. Downey, N. Greenberg, A.E.M. Lewis, A. Montalbán, Extensions of uppersemilattice embeddings below computably enumerable degrees, submitted.
[MG]: M. Giorgi, PhD thesis, Leeds University.
[GMS]: N. Greenberg, A. Montalbán, R. Shore, Generalized high degrees have the complementation property, Journal of Symbolic Logic, 69 (2004), 1200-1220. MR 2135663 (2006a:03060)
[CJ]: C. Jockusch, Simple proofs of some theorems on high degrees of unsolvability, Canadian Journal of Mathematics, 29 (1977), 1072-1080. MR 0476460 (57:16023)
[JP]: C. Jockusch, D. Posner, Double jumps of minimal degrees, Journal of Symbolic Logic, 43 (4) (1978), 715-724. MR 518677 (80d:03042)
[AK]: A. Kučera, Measure, $\Pi^0_1$ -classes and complete extensions of PA, Recursion Theory Week (Oberwolfach, 1984), volume 1141 of Lecture Notes in Math., 245-259, Springer, Berlin, 1985. MR 820784 (87e:03102)
[ML]: M. Lerman, Degrees of Unsolvability, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. MR 708718 (85h:03044)
[AL]: A.E.M. Lewis, Minimal complements for degrees below $\boldsymbol{0}'$ , Journal of Symbolic Logic, 69 (4) (2004), 937-966. MR 2135652 (2005m:03087)
[AL2]: A.E.M. Lewis, The minimal complementation property above $\boldsymbol{0}'$ , Mathematical Logic Quarterly, 51 (5) (2005), 470-492. MR 2163759 (2006d:03066)
[DP]: D. Posner, A survey of the non-r.e. degrees $\leq \boldsymbol{0}'$ , London Mathematical Society Lecture Note Series, 45, Recursion Theory: Its Generalisations and Applications, Proceedings of the Logic Colloquium '79, Leeds, August 1979, edited by F.R. Drake and S.S. Wainer, Cambridge Univ Press, Cambridge-New York, 1980. MR 598301 (81j:03007)
[GS]: G. Sacks, A minimal degree less than $\boldsymbol{0}'$ , Bulletin of the American Mathematical Society 67 (1961), 416-419. MR 0126380 (23:A3676)
[RS]: R.I. Soare, Recursively Enumerable Sets and Degrees, Springer, New York, 1987. MR 882921 (88m:03003)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03D28, 03D10

Retrieve articles in all Journals with MSC (2000): 03D28, 03D10

Additional Information:

Philip Ellison
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds, LS29JT, England
Email: phil.j.ellison@googlemail.com

Andrew E. M. Lewis
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds, LS29JT, England
Email: andy@aemlewis.co.uk

DOI: 10.1090/S0002-9939-10-10299-8
PII: S 0002-9939(10)10299-8
Received by editor(s): March 8, 2009
Received by editor(s) in revised form: September 20, 2009 and November 20, 2009
Posted: March 29, 2010
Additional Notes: The first author was supported by an EPSRC research studentship.
The second author was supported by a Royal Society University Research Fellowship
Communicated by: Julia Knight
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|