On a problem of Ishmukhametov

• Main Authors: Yamaleev, Mars, Fang, Chengling, Wu, Guohua
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2014
• Subjects: DRNTU::Science::Mathematics
• Online Access: https://hdl.handle.net/10356/103046; http://hdl.handle.net/10220/19230
• Institution: Nanyang Technological University

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.