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

  • Authors:

  • Douglas Cenzer;Johanna N. Franklin;Jiang Liu;Guohua Wu

  • Affiliations:

  • Department of Mathematics, University of Florida, Gainesville, USA 32611-8105;Department of Mathematics, 6188 Kemeny Hall, Dartmouth College, Hanover, USA 03755-3551;Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore 637371;Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore 637371

  • Venue:
  • Archive for Mathematical Logic
  • Year:
  • 2011

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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.