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Archive for Mathematical Logic

, Volume 38, Issue 6, pp 373–386 | Cite as

On the r.e. predecessors of d.r.e. degrees

  • Shamil Ishmukhametov

Abstract.

Let d be a Turing degree containing differences of recursively enumerable sets (d.r.e.sets) and R[d] be the class of less than d r.e. degrees in whichd is relatively enumerable (r.e.). A.H.Lachlan proved that for any non-recursive d.r.e. d R[d] is not empty. We show that the r.e. degree defined by Lachlan for a d.r.e.set \(D\in\) d is just the minimum degree in which D is r.e. Then we study for a given d.r.e. degree d class R[d] and show that there exists a d.r.e.d such that R d] has a minimum element \(>\) 0. The most striking result of the paper is the existence of d.r.e. degrees for which R[d] consists of one element. Finally we prove that for some d.r.e. d R[d] can be the interval [a,b] for some r.e. degrees a,b, a \(<\) b \(<\) d.

Keywords

Minimum Degree Minimum Element Striking Result Turing Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Shamil Ishmukhametov
    • 1
  1. 1. Department of Mathematics, Tolstoy St. 42, Ulyanovsk University, Ulyanovsk, Russia, 432700. E-mail: ishm@mci.univ.simbirsk.su RU

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