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

  • Authors:

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

  • Affiliations:

  • Department of Mathematics, University of Florida, Gainesville, USA FL 32611-8105;Department of Mathematics, National University of Singapore 2, Singapore, Singapore 117543;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:
  • TAMC '09 Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation
  • Year:
  • 2009

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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.