Embedding the Diamond Lattice in the c.e. tt-Degrees with Superhigh Atoms

  • Douglas Cenzer
  • Johanna N. Y. Franklin
  • Jiang Liu
  • Guohua Wu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)


The notion of superhigh computably enumerable (c.e.) degrees was first introduced by Mohrherr in [7], where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to algorithmic randomness and effective measure theory. Jockusch and Mohrherr proved in [4] that the diamond lattice can be embedded into the c.e. tt-degrees preserving 0 and 1 and that the two atoms can be low. In this paper, we prove that the two atoms in such embeddings can also be superhigh.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cooper, S.B.: Degrees of unsolvability complementary between recursively enumerable degrees. I. Ann. Math. Logic 4, 31–73 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Downey, R.: D.r.e. degrees and the nondiamond theorem. Bull. London Math. Soc. 21, 43–50 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Epstein, R.L.: Minimal degrees of unsolvability and the full approximation construction. Mem. Amer. Math. Soc. 162(3) (1975)Google Scholar
  4. 4.
    Jockusch Jr., C.G., Mohrherr, J.: Embedding the diamond lattice in the recursively enumerable truth-table degrees. Proc. Amer. Math. Soc. 94, 123–128 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lachlan, A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16, 537–569 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Martin, D.: Classes of recursively enumerable sets and degrees of unsolvability. Z. Math. Logik Grundlag. Math. 12, 295–310 (1966)zbMATHCrossRefGoogle Scholar
  7. 7.
    Mohrherr, J.: A refinement of lown and highn for the r.e. degrees. Z. Math. Logik Grundlag. Math. 32, 5–12 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ng, K.M.: On Very High Degrees. Jour. Symb. Logic 73, 309–342 (2008)zbMATHCrossRefGoogle Scholar
  9. 9.
    Simpson, S.G.: Almost everywhere domination and superhighness. Math. Log. Q. 53, 462–482 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Soare, R.I.: Recursively enumerable sets and degrees. Springer, Heidelberg (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Douglas Cenzer
    • 1
  • Johanna N. Y. Franklin
    • 2
  • Jiang Liu
    • 3
  • Guohua Wu
    • 3
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsNational University of Singapore 2SingaporeSingapore
  3. 3.Division of Mathematical Sciences School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

Personalised recommendations