An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees
Klaus Ambos‐Spies, Carl G. Jockusch, Richard A. Shore, Robert Irving Soare · Transactions of the American Mathematical Society · 1984
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