An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees

Author:

Ambos-Spies Klaus,Jockusch Carl G.,Shore Richard A.,Soare Robert I.

Abstract

We specify a definable decomposition of the upper semilattice of recursively enumerable (r.e.) degrees R \mathbf {R} as the disjoint union of an ideal M \mathbf {M} and a strong filter N C \mathbf {NC} . The ideal M \mathbf {M} consists of 0 \mathbf {0} together with all degrees which are parts of r.e. minimal pairs, and thus the degrees in N C \mathbf {NC} are called noncappable degrees. Furthermore, N C \mathbf {NC} coincides with five other apparently unrelated subclasses of R : E N C \mathbf {R: ENC} , the effectively noncappable degrees; P S \mathbf {PS} , the degrees of promptly simple sets; L C \mathbf {LC} , the r.e. degrees cuppable to 0 {\mathbf {0}}’ by a low r.e. degree; S P H ¯ {\mathbf {SP\bar H}} , the degrees of non- h h hh -simple r.e. sets with the splitting property; and G \mathbf {G} , the degrees in the orbit of an r.e. generic set under automorphisms of the lattice of r.e. sets.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference28 articles.

1. K. Ambos-Spies, On the structure of the recursively enumerable degrees, Ph.D. dissertation, Univ. of Munich.

2. \bysame, An extension of the non-diamond theorem in classical and 𝛼-recursion theory, J. Symbolic Logic.

3. \bysame, Non-cappability in the r.e. weak truth table and Turing degrees,

4. P. Fejer, The structure of definable subclasses of the recursively enumerable degrees, Ph.D. dissertation, Univ. of Chicago.

5. The plus-cupping theorem for the recursively enumerable degrees;Fejer, Peter A.,1981

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