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On a problem of Ishmukhametov

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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.

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Authors and Affiliations

  1. School of Science, Chongqing Jiaotong University, Chongqing, 400074, People’s Republic of China

    Chengling Fang

  2. School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Guohua Wu

  3. Institute of Mathematics and Mechanics, Kazan Federal University, 18 Kremlyovskaya Str., Kazan, 420008, Russia

    Mars Yamaleev

Authors
  1. Chengling Fang
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  2. Guohua Wu
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  3. Mars Yamaleev
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Corresponding author

Correspondence to Guohua Wu.

Additional information

Fang is supported by Chongqing Jiaotong University Fund (No. 2012kjc2-018).

Wu is supported in part by NTU grant RG37/09, M52110101 and grant MOE2011-T2-1-071 from Ministry of Education of Singapore.

Yamaleev is supported by NTU grant RG37/09, M52110101, by Russian Foundation for Basic Research (projects RFBR-12-01-97008, RFBR-12-01-31389), by The President grant of Russian Federation (project SS-5383.2012.1), and by The Ministry of education and science of Russian Federation (projects 14.A18.21.0360, 14.A18.21.0368, 14.A18.21.1127).

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Fang, C., Wu, G. & Yamaleev, M. On a problem of Ishmukhametov. Arch. Math. Logic 52, 733–741 (2013). https://doi.org/10.1007/s00153-013-0340-0

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  • DOI: https://doi.org/10.1007/s00153-013-0340-0

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