Archive for Mathematical Logic 49 (1):35-49 (2010)

Lachlan observed that the infimum of two r.e. degrees considered in the r.e. degrees coincides with the one considered in the ${\Delta_2^0}$ degrees. It is not true anymore for the d.r.e. degrees. Kaddah proved in (Ann Pure Appl Log 62(3):207–263, 1993) that there are d.r.e. degrees a, b, c and a 3-r.e. degree x such that a is the infimum of b, c in the d.r.e. degrees, but not in the 3-r.e. degrees, as a < x < b, c. In this paper, we extend Kaddah’s result by showing that such a structural difference occurs densely in the r.e. degrees. Our result immediately implies that the isolated 3-r.e. degrees are dense in the r.e. degrees, which was first proved by LaForte
Keywords Turing degrees  infimum  Ershov hierarchy
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Reprint years 2009, 2010
DOI 10.1007/s00153-009-0159-x
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References found in this work BETA

Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers.Piergiorgio Odifreddi - 1989 - Sole Distributors for the Usa and Canada, Elsevier Science Pub. Co..
Infima in the D.R.E. Degrees.D. Kaddah - 1993 - Annals of Pure and Applied Logic 62 (3):207-263.
Isolation in the CEA Hierarchy.Geoffrey LaForte - 2005 - Archive for Mathematical Logic 44 (2):227-244.

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