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Journal of Logic and Computation Advance Access originally published online on August 2, 2007
Journal of Logic and Computation 2007 17(6):1153-1166; doi:10.1093/logcom/exm039
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Articles

Isolation, Infima and Diamond Embeddings

Jiang Liu

School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639798, Republic of Singapore. E-mail: LIUJ0027{at}ntu.edu.sg

Guohua Wu *

School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639798, Republic of Singapore. E-mail: guohua{at}ntu.edu.sg

Received 16 October 2006.


  Abstract

In (1993, Annals of Pure and Applied Logic, 62, 207–263), Kaddah pointed out that there are two d.c.e. degrees a,b forming a minimal pair in the d.c.e. degrees, but not in the Formula degrees. Kaddah's; result shows that Lachlan's; theorem, stating that the infima of two c.e. degrees in the c.e. degrees and in the Formula degrees coincide, cannot be generalized to the d.c.e. degrees.

In this article, we apply Kaddah's; idea to show that there are two d.c.e. degrees c,d such that c cups d to 0', and caps d to 0 in the d.c.e. degrees, but not in the Formula degrees. As a consequence, the diamond embedding {0,c,d,0'} is different from the one first constructed by Downey in 1989 in [5]. To obtain this, we will construct c.e. degrees a,b, d.c.e. degrees c > a,d > b and a non-zero {omega}-c.e. degree e ≤ c,d such that (i) a,b form a minimal pair, (ii) a isolates c, and (iii) b isolates d. From this, we can have that c,d form a minimal pair in the d.c.e. degrees, and Kaddah's; result follows immediately. In our construction, we apply Kaddah's; original idea to make e below both c and d. Our construction allows us to separate the minimal pair argument (a{cap}b = 0), the splitting of 0' (c{cup}d = 0'), and the non-minimal pair of c,d (in the Formula degrees), into several parts, to avoid direct conflicts that could be involved if only c,d and e are constructed. We also point out that our construction allows us to make a,b above (and hence c,d) high.

Keywords: Turing degrees; isolation; diamond embeddings

*Partially supported by a start-up grant No. M48110008 and a research grant No. RG58/06 from NTU.


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