On a problem of Ishmukhametov

  • Authors:

  • Chengling Fang;Guohua Wu;Mars Yamaleev

  • Affiliations:

  • School of Science, Chongqing Jiaotong University, Chongqing, People's Republic of China 400074;School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore 637371;Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russia 420008

  • Venue:
  • Archive for Mathematical Logic
  • Year:
  • 2013

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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.