TY - JOUR
AU - Wu, Guohua
AB - 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 Δ20 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 Δ20 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 Δ20 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 ω-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∩b = 0), the splitting of 0′ (c∪d = 0'), and the non-minimal pair of c,d (in the Δ20 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.
TI - Isolation, Infima and Diamond Embeddings
JF - Journal of Logic and Computation
DO - 10.1093/logcom/exm039
DA - 2007-12-02
UR - https://www.deepdyve.com/lp/oxford-university-press/isolation-infima-and-diamond-embeddings-6fVvgPgHmi
SP - 1153
EP - 1166
VL - 17
IS - 6
DP - DeepDyve
ER -