Source: J. Symbolic Logic
Volume 69, Issue 2
In this paper, we study the bi-isolation phenomena in the d.c.e.
degrees and prove that there are c.e. degrees c1
< c2 and a d.c.e. degree d∈ (c1, c2) such that
(c1, d) and
(d, c2) contain no c.e.
degrees. Thus, the c.e. degrees between c1 and
c2 are all incomparable with d. We also
show that there are d.c.e. degrees d1 <
d2 such that
(d1, d2) contains a unique c.e. degree.
M. M. Arslanov Structural properties of the degrees below $\textup\bfseries 0'$, Doklady Akademii Nauk UzSSR, vol. 283 (1985), pp. 270--273.
Mathematical Reviews (MathSciNet): MR804113
M. M. Arslanov, S. Lempp, and R. A. Shore On isolating r.e. and isolated d.r.e. degrees, Computability, enumerability, unsolvability (S. B. Cooper, T. A. Slaman, and S. S. Wainer, editors),1996, pp. 61--80.
S. B. Cooper A splitting theorem of the $n$-r.e. degrees, Proceedings of the American Mathematical Society, vol. 115, pp. 461--471.
S. B. Cooper, L. Harrington, A. H. Lachlan, S. Lempp, and R. I. Soare The d.r.e. degrees are not dense, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 125--151.
S. B. Cooper, S. Lempp, and P. Watson Weak density and cupping in the $d$-r.e. degrees, Israel Journal of Mathematics, vol. 67 (1989), pp. 137--152.
S. B. Cooper and X. Yi Isolated d.r.e. degrees, \textnormalpreprint series \textnormal17, University of Leeds, Department of Pure Mathematics,1995, 25 pp.
D. Ding and L. Qian Isolated d.r.e. degrees are dense in r.e. degree structure, Archive for Mathematical Logic, vol. 36 (1996), pp. 1--10.
R. G. Downey D.r.e. degrees and the nondiamond theorem, Bulletin of the London Mathematical Society, vol. 21 (1989), pp. 43--50.
Mathematical Reviews (MathSciNet): MR967789
A. Efremov Upper isolated d.c.e. degrees, I, Izvestiya Vysshikh Uchebnykh Zavedeniĭ, Matematika, vol. 42 (1998), no. 2, pp. 20--28, in Russian.
Yu. L. Ershov On a hierarchy of sets I, Algebra i Logika, vol. 7 (1968), pp. 47--73.
Mathematical Reviews (MathSciNet): MR270911
L. Harrington and R. I. Soare Games in recursion theory and continuity properties of capping degrees, Set theory and the continuum (H. Judah, W. Just, and W. H. Woodin, editors),1992, pp. 39--62.
S. Ishmukhametov D.r.e. sets, their degrees and index sets, Thesis, Novosibirsk, Russia,1986.
S. Ishmukhametov and G. Wu Isolation and the high/low hierarchy, Archive for Mathematical Logic, vol. 41 (2002), pp. 259--266.
G. LaForte The isolated d.r.e. degrees are dense in the r.e. degrees, Mathematical Logic Quarterly, vol. 42 (1996), pp. 83--103.
S. Lempp private communications, April 2001.
A. Li and X. Yi Cupping the recursively enumerable degrees by d.r.e. degrees, Proceedings of the London Mathematical Society, vol. 78 (1999), pp. 1--21.
G. E. Sacks On the degrees less than $\textup\bfseries 0'$, Annals of Mathematics, vol. 77 (1963), pp. 211--231.
Mathematical Reviews (MathSciNet): MR146078
R. I. Soare Recursively enumerable sets and degrees, Springer-Verlag, Berlin,1987.
Mathematical Reviews (MathSciNet): MR882921
G. Wu Isolation and the jump operator, Mathematical Logic Quarterly, vol. 47 (2001), pp. 525--534.