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A superhigh diamond in the c.e. tt-degrees

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Abstract

The notion of superhigh computably enumerable (c.e.) degrees was first introduced by (Mohrherr in Z Math Logik Grundlag Math 32: 5–12, 1986) 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 (Proc Amer Math Soc 94:123–128, 1985) 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. Lond. 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)

  4. Jockusch C.G. Jr, Mohrherr J.: Embedding the diamond lattice in the recursively enumerable truth-table degrees. Proc. Amer. Math. Soc. 94, 123–128 (1985)

    MATH  MathSciNet  Google Scholar 

  5. Lachlan A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. Lond. 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. J. 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, 6188 Kemeny Hall, Dartmouth College, Hanover, NH, 03755-3551, USA

    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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Corresponding author

Correspondence to Jiang Liu.

Additional information

Cenzer is partially supported by NSF grants DMS-0554841 and DMS-652372.

Wu is partially supported by AcRF grants RG58/06 and RG37/09 from Singapore MoE.

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Cenzer, D., Franklin, J.N.Y., Liu, J. et al. A superhigh diamond in the c.e. tt-degrees. Arch. Math. Logic 50, 33–44 (2011). https://doi.org/10.1007/s00153-010-0198-3

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  • DOI: https://doi.org/10.1007/s00153-010-0198-3

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