Abstract
In this paper, we prove the existence of a minimal pair of c.e. degrees a and b such that both of them are cuppable, and no incomplete c.e. degree can cup both of them to 0′. As a consequence, [a] and [b] form a minimal pair in M/NCup, the quotient structure of the cappable degrees modulo noncuppable degrees. We also prove that the dual of Lempp’s conjecture is true.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ambos-Spies, K., Jockusch Jr., C.G., Shore, R.A., Soare, R.I.: An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees. Trans. Amer. Math. Soc. 281, 109–128 (1984)
Cooper, S.B.: Minimal pairs and high recursively enumerable degrees. J. Symbolic Logic 39, 655–660 (1974)
Cooper, S.B.: On a theorem of C. E. M. Yates. Handwritten notes (1974)
Cooper, S.B.: Computability Theory, Chapman & Hall/CRC Mathematics, Boca Raton, FL, New York, London (2004)
Downey, R., Lempp, S.: There is no plus-capping degree. Arch. Math. Logic 33, 109–119 (1994)
Fejer, P.A., Soare, R.I.: The plus-cupping theorem for the recursively enumerable degrees. In: Logic Year 1979–80: University of Connecticut, pp. 49–62 (1981)
Lachlan, A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16, 537–569 (1966)
Lachlan, A.H.: Bounding minimal pairs. J. Symb. Logic 44, 626–642 (1979)
Li, A.: On a conjecture of Lempp. Arch. Math. Logic 39, 281–309 (2000)
Li, A., Wu, G., Yang, Y.: On the quotient structure of computably enumerable degrees modulo the noncuppable ideal. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 731–736. Springer, Heidelberg (2006)
Li, A., Wu, G., Yang, Y.: Embed the diamond lattice into R/NCup preserving 0 and 1. In preparation
Miller, D.: High recursively enumerable degrees and the anticupping property. In: Logic Year 1979–80: University of Connecticut, pp. 230–245 (1981)
Sacks, G.E.: On the degrees less than 0’. Ann. of Math. 77, 211–231 (1963)
Sacks, G.E.: The recursively enumerable degrees are dense. Ann. of Math. 80, 300–312 (1964)
Schwarz, S.: The quotient semilattice of the recursively enumerable degrees modulo the cappable degrees. Trans. Amer. Math. Soc. 283, 315–328 (1984)
Shoenfield, J.R.: Applications of model theory to degrees of unsolvability. In: Addison, J.W., Henkin, L., Tarski, A. (eds.) The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley. Studies in Logic and the Foundations of Mathematics, pp. 359–363. Holland Publishing, Amsterdam (1965)
Slaman, T.: Questions in recursion theory. In: Cooper, S.B., Slaman, T.A., Wainer, S.S. (eds.) Computability, enumerability, unsolvability. Directions in recursion theory. London Mathematical Society Lecture Note Series, vol. 224, pp. 333–347. Cambridge University Press, Cambridge (1996)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Perspective in Mathematical Logic, Springer-Verlag, Berlin, Heidelberg, New York
Sui, Y., Zhang, Z.: The cupping theorem in R/ M. J. Symbolic Logic 64, 643–650 (1999)
Yates, C.E.M.: A minimal pair of recursively enumerable degrees. J. Symbolic Logic 31, 159–168 (1966)
Yates, C.E.M.: On the degrees of index sets. Trans. Amer. Math. Soc. 121, 309–328 (1966)
Yi, X.: Extension of embeddings on the recursively enumerable degrees modulo the cappable degrees. In: Computability, enumerability, unsolvability, Directions in recursion theory (eds. Cooper, Slaman, Wainer), London Mathematical Society Lecture Note Series 224, pp. 313–331
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bie, R., Wu, G. (2007). A Minimal Pair in the Quotient Structure M/NCup . In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-73001-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73000-2
Online ISBN: 978-3-540-73001-9
eBook Packages: Computer ScienceComputer Science (R0)