Archive for Mathematical Logic 52 (7-8):733-741 (2013)

Abstract
Given a d.c.e. degree d, consider the d.c.e. sets in d and the corresponding degrees of their Lachlan sets. Ishmukhametov provided a systematic investigation of such degrees, and proved that for a given d.c.e. degree d > 0, the class of its c.e. predecessors in which d is c.e., denoted as R[d], can consist of either just one element, or an interval of c.e. degrees. After this, Ishmukhametov asked whether there exists a d.c.e. degree d for which the class R[d] has no minimal element. We give a positive answer to this question
Keywords d.c.e. sets  Lachlan sets  Turing degrees
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DOI 10.1007/s00153-013-0340-0
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On the R.E. Predecessors of D.R.E. Degrees.Shamil Ishmukhametov - 1999 - Archive for Mathematical Logic 38 (6):373-386.

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