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Degree theoretical splitting properties of recursively enumerable sets

Klaus Ambos-Spies & Peter A. Fejer

Journal of Symbolic Logic 53 (4):1110-1137 (1988)

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Embeddings of N5 and the contiguous degrees.Klaus Ambos-Spies & Peter A. Fejer - 2001 - Annals of Pure and Applied Logic 112 (2-3):151-188.details Downey and Lempp 1215–1240) have shown that the contiguous computably enumerable degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truth-table degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5-element lattice N5 into the c.e. degrees which maps the top to the degree a. In particular, this shows that local nondistributivity coincides with (...) Download Export citation Bookmark 4 citations
Parameter definability in the recursively enumerable degrees.André Nies - 2003 - Journal of Mathematical Logic 3 (01):37-65.details The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the [Formula: see text] relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the [Formula: see text] relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low 1 is parameter definable, and we provide methods that lead to a new example of a ∅-definable ideal. Moreover, we prove that (...) Download Export citation Bookmark 6 citations
Jumps of nontrivial splittings of recursively enumerable sets.Michael A. Ingrassia & Steffen Lempp - 1990 - Mathematical Logic Quarterly 36 (4):285-292.details Download Export citation Bookmark 2 citations
Completely mitotic c.e. degrees and non-jump inversion.Evan J. Griffiths - 2005 - Annals of Pure and Applied Logic 132 (2-3):181-207.details A completely mitotic computably enumerable degree is a c.e. degree in which every c.e. set is mitotic, or equivalently in which every c.e. set is autoreducible. There are known to be low, low2, and high completely mitotic degrees, though the degrees containing non-mitotic sets are dense in the c.e. degrees. We show that there exists an upper cone of c.e. degrees each of which contains a non-mitotic set, and that the completely mitotic c.e. degrees are nowhere dense in the c.e. (...) Download Export citation Bookmark
Splitting theorems and the jump operator.R. G. Downey & Richard A. Shore - 1998 - Annals of Pure and Applied Logic 94 (1-3):45-52.details We investigate the relationship of the degrees of splittings of a computably enumerable set and the degree of the set. We prove that there is a high computably enumerable set whose only proper splittings are low 2. Download Export citation Bookmark 1 citation
Splitting theorems in recursion theory.Rod Downey & Michael Stob - 1993 - Annals of Pure and Applied Logic 65 (1):1-106.details A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...) Download Export citation Bookmark 18 citations
Completely mitotic R.E. degrees.R. G. Downey & T. A. Slaman - 1989 - Annals of Pure and Applied Logic 41 (2):119-152.details Download Export citation Bookmark 10 citations

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