Nonexistence of Minimal Pairs in $L[{\mathbf d}]$

Abstract

For a d.c.e. set $D$ with a d.c.e. approximation $\{D_s\}_{s\in \omega }$, the Lachlan set of $D$ is defined as $L(D) = \{ s: \exists x \in D_{s} - D_{s-1} \ \hbox {and} \ x \not \in D\}.$ For a d.c.e. degree ${\mathbf d}$, $L[\mathbf{d}]$ is defined as the class of c.e. degrees of those Lachlan sets of d.c.e. sets in $\mathbf{d}$. In this paper, we prove that for any proper d.c.e. degree $\mathbf{d}$, no two elements in $L[\mathbf{d}]$ can form a minimal pair. This result gives another solution to Ishmukhametov’s problem, which asks whether for any proper d.c.e. degree $\mathbf{d}$, $L[\mathbf{d}]$ always has a minimal element. A negative answer to this question was first given by Fang, Wu and Yamaleev in 2013.

Fang is partially supported by NSF of China (No. 11401061), SRF for ROCS, SEM and Chongqing Jiaotong University Fund (No. 2012kjc2-018).

Liu is partially supported by NSF of China (No. 61202131), the CAS western light program, Chongqing Natural Science Foundation (No. cstc2014jcsfglyjs0005 and No. cstc2014zktjccxyyB0031).

Wu is partially supported by a grant MOE2011-T2-1-071 (ARC 17/11, M45110030) from Ministry of Education of Singapore and by a grant RG29/14, M4011274 from NTU.

Yamaleev is partially supported by The President grant of Russian Federation (project NSh-941.2014.1), by Russian Foundation for Basic Research (projects 14-01-31200, 15-01-08252), by the subsidy allocated to Kazan Federal University for the project part of the state assignment in the sphere of scientific activities, and by a grant MOE2011-T2-1-071 (ARC 17/11, M45110030) from Ministry of Education of Singapore.