Isolation, Infima and Diamond Embeddings

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.