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There Are No Maximal Low D.C.E. Degrees

Liang Yu & Rod Downey

Notre Dame Journal of Formal Logic 45 (3):147-159 (2004)

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There is No Low Maximal D.C.E. Degree.Marat Arslanov, S. Barry Cooper & Angsheng Li - 2000 - Mathematical Logic Quarterly 46 (3):409-416.details We show that for any computably enumerable set A and any equation image set L, if L is low and equation image, then there is a c.e. splitting equation image such that equation image. In Particular, if L is low and n-c.e., then equation image is n-c.e. and hence there is no low maximal n-c.e. degree. Download Export citation Bookmark 3 citations
Gerald E. Sacks. The recursively enumerable degrees are dense. Annals of mathematics, ser. 2 vol. 80 (1964), pp. 300–312. [REVIEW]Gerald E. Sacks - 1969 - Journal of Symbolic Logic 34 (2):294-295.details Download Export citation Bookmark 38 citations
The d.r.e. degrees are not dense.S. Barry Cooper, Leo Harrington, Alistair H. Lachlan, Steffen Lempp & Robert I. Soare - 1991 - Annals of Pure and Applied Logic 55 (2):125-151.details By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees. Download Export citation Bookmark 31 citations
The d.r.e. degrees are not dense.S. Cooper, Leo Harrington, Alistair Lachlan, Steffen Lempp & Robert Soare - 1991 - Annals of Pure and Applied Logic 55 (2):125-151.details By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees. Download Export citation Bookmark 24 citations
Relative enumerability in the difference hierarchy.Marat Arslanov, Geoffrey Laforte & Theodore Slaman - 1998 - Journal of Symbolic Logic 63 (2):411-420.details We show that the intersection of the class of 2-REA degrees with that of the ω-r.e. degrees consists precisely of the class of d.r.e. degrees. We also include some applications and show that there is no natural generalization of this result to higher levels of the REA hierarchy. Download Export citation Bookmark 6 citations
A recursively enumerable degree which will not split over all lesser ones.Alistair H. Lachlan - 1976 - Annals of Mathematical Logic 9 (4):307.details Download Export citation Bookmark 39 citations

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