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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Klaus Ambos-Spies, Carl G. Jockusch, Richard A. Shore and Robert I. Soare
Journal: Trans. Amer. Math. Soc. 281 (1984), 109-128
MSC: Primary 03D25; Secondary 03D30
MathSciNet review: 719661
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Abstract: We specify a definable decomposition of the upper semilattice of recursively enumerable (r.e.) degrees $ \mathbf{R}$ as the disjoint union of an ideal $ \mathbf{M}$ and a strong filter $ \mathbf{NC}$. The ideal $ \mathbf{M}$ consists of $ \mathbf{0}$ together with all degrees which are parts of r.e. minimal pairs, and thus the degrees in $ \mathbf{NC}$ are called noncappable degrees. Furthermore, $ \mathbf{NC}$ coincides with five other apparently unrelated subclasses of $ \mathbf{R: ENC}$, the effectively noncappable degrees; $ \mathbf{PS}$, the degrees of promptly simple sets; $ \mathbf{LC}$, the r.e. degrees cuppable to $ {\mathbf{0}}^{\prime}$ by a low r.e. degree; $ {\mathbf{SP\bar H}}$, the degrees of non-$ hh$-simple r.e. sets with the splitting property; and $ \mathbf{G}$, the degrees in the orbit of an r.e. generic set under automorphisms of the lattice of r.e. sets.

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Article copyright: © Copyright 1984 American Mathematical Society