An Algebraic Decomposition of the Recursively Enumerable Degrees and the Coincidence of Several Degree Classes with the Promptly Simple Degrees1

We specify a definable decomposition of the upper semilattice of recursively enumerable (r.e.) degrees R as the disjoint union of an ideal M and a strong filter NC. The ideal M consists of 0 together with all degrees which are parts of r.e. minimal pairs, and thus the degrees in NC are called noncappable degrees. Furthermore, NC coincides with five other apparently unrelated subclasses of R: ENC, the effectively noncappable degrees; PS, the degrees of promptly simple sets; LC, the r.e. degrees cuppable to 0' by a low r.e. degree; SPH, the degrees of non-Wi-simple r.e. sets with the splitting property; and G, the degrees in the orbit of an r.e. generic set under automorphisms of the lattice of r.e. sets. 0. Introduction. Let (R, < , U, n) denote the upper semilattice of recursively enumerable (r.e.) degrees with partial ordering induced by Turing reducibility and U and n the join and meet operations when the latter is defined. (Unless otherwise specified all sets and degrees will be assumed to be r.e.) Sacks [1966, p. 170] asked whether there exists a minimal pair namely incomparable r.e. degrees a and b such that a n b = 0. Shoenfield [1965] formulated a general conjecture about R which implies among other things that minimal pairs do not exist. Lachlan [1966] and independently Yates [1966] refuted Shoenfield's conjecture by constructing a minimal pair. Both minimal pairs and the method for constructing them have played an important role in the study of r.e. degrees. An r.e. degree a is cappable (caps) if there is an r.e. degree b > 0 such that a n b = 0 (i.e. if a is 0 or is part of a minimal pair), and a is noncappable otherwise. Furthermore, a is effectively noncappable if the witness to its noncapping can be found effectively (as defined more precisely in §1). Yates [1966] also showed that there exist r.e. degrees a < 0' which are noncappable, indeed effectively noncappable. Let M, NC and ENC denote the classes of cappable, noncappable and effectively noncappable r.e. degrees, respectively. We prove that M is an ideal in R (closed downward and under join) while its complement NC is a strong filter (closed upwards and for all a, b e NC there exists Received by the editors June 14, 1982. 1980 Mathematics Subject Classification. Primary 03D25; Secondary 03D30. 'The last three authors were partially supported by NSF Grants MCS-8003251, MCS-8003016, MCS-7905782, and MCS-8102443 respectively. ©1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page