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

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Theory and Applications of Models of Computation (TAMC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5532))

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Abstract

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.

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References

  1. Cooper, S.B.: Degrees of unsolvability complementary between recursively enumerable degrees. I. Ann. Math. Logic 4, 31–73 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Downey, R.: D.r.e. degrees and the nondiamond theorem. Bull. London Math. Soc. 21, 43–50 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Epstein, R.L.: Minimal degrees of unsolvability and the full approximation construction. Mem. Amer. Math. Soc. 162(3) (1975)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lachlan, A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16, 537–569 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  6. Martin, D.: Classes of recursively enumerable sets and degrees of unsolvability. Z. Math. Logik Grundlag. Math. 12, 295–310 (1966)

    Article  MATH  Google Scholar 

  7. Mohrherr, J.: A refinement of low n and high n for the r.e. degrees. Z. Math. Logik Grundlag. Math. 32, 5–12 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ng, K.M.: On Very High Degrees. Jour. Symb. Logic 73, 309–342 (2008)

    Article  MATH  Google Scholar 

  9. Simpson, S.G.: Almost everywhere domination and superhighness. Math. Log. Q. 53, 462–482 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Soare, R.I.: Recursively enumerable sets and degrees. Springer, Heidelberg (1987)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, 310 Little Hall, Gainesville, FL 32611-8105, USA

    Douglas Cenzer

  2. Department of Mathematics, National University of Singapore 2, Science Drive 2, Singapore, 117543, Singapore

    Johanna N. Y. Franklin

  3. Division of Mathematical Sciences School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Jiang Liu & Guohua Wu

Authors
  1. Douglas Cenzer
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  2. Johanna N. Y. Franklin
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  3. Jiang Liu
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  4. Guohua Wu
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Texas A&M University, College Station, 77843, Texas, USA

    Jianer Chen

  2. School of Mathematics, University of Leeds, LS2 9JT, U.K.

    S. Barry Cooper

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© 2009 Springer-Verlag Berlin Heidelberg

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Cenzer, D., Franklin, J.N.Y., Liu, J., Wu, G. (2009). Embedding the Diamond Lattice in the c.e. tt-Degrees with Superhigh Atoms. In: Chen, J., Cooper, S.B. (eds) Theory and Applications of Models of Computation. TAMC 2009. Lecture Notes in Computer Science, vol 5532. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02017-9_44

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  • DOI: https://doi.org/10.1007/978-3-642-02017-9_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02016-2

  • Online ISBN: 978-3-642-02017-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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