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Archive for Mathematical Logic

, Volume 50, Issue 1–2, pp 33–44 | Cite as

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

  • Douglas Cenzer
  • Johanna N. Y. Franklin
  • Jiang LiuEmail author
  • Guohua Wu
Article
  • 37 Downloads

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.

Keywords

Computably enumerable sets Truth-table degrees Superhighness Lattice embeddings 

Mathematics Subject Classification (2000)

Primary 03D25 Secondary 03D30 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Douglas Cenzer
    • 1
  • Johanna N. Y. Franklin
    • 2
  • Jiang Liu
    • 3
    Email author
  • Guohua Wu
    • 3
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of Mathematics, 6188 Kemeny HallDartmouth CollegeHanoverUSA
  3. 3.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

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