The Quotient Semilattice of the Recursively Enumerable Degrees modulo the Cappable

In this paper, we investigate the quotient semilattice R/M of the r.e. degrees modulo the cappable degrees. We first prove the R/M counterpart of the Friedberg-Muchnik theorem. We then show that minimal elements and minimal pairs are not present in R/M. We end with a proof of the R/M counterpart to Sack's splitting theorem. 0. Introduction. The set of all r.e. degrees is made into an upper semilattice (with 0 and 1) in a natural way: namely, the reducibility relation between r.e. sets induces a partial ordering on degrees, for which it is readily shown that finite suprema always exist. This semilattice structure, denoted "7 ", has been extensively studied. Earliest results stress the richness and the uniformity of 7. For instance, the Friedberg-Muchnik theorem states that there exists an incomparable pair in 7 (Friedberg [1957] and Muchnik [1956]). This may be extended to obtain the existence of a countably infinite independent set in 7. It then follows that any countable partial ordering may be embedded into 7 (Sacks [1966]). Another example: Sacks' splitting theorem states that any element of 7 may be written as the supremum of an imcomparable pair. In fact, such a pair may always be chosen to lie outside any preassigned (nontrivial) principal filter (Sacks [1963]). A corollary of this is that every element of 7 (except the least and greatest elements) is half of an incomparable pair. The culmination of the early uniformity results about 7 is Sacks' density theorem, which states that, whenever a and b are elements of R such that a < b, then c can be found in 7 such that a < c < b (Sacks [1964]). J. Shoenfield [1965] responded to this result by conjecturing its strongest possible generalization: namely, that whenever a,...,a„ satisfy a diagram D = D(xx,...,xn) in 7 and Dl = Dx(xx,...,xn, y) is a diagram which xtends D, and which is suitably consistent (i.e. it actually occurs in some upper semilattice with 0 and 1), then b can be found in 7 such that ax,...,a„, b satisfy Dx in 7. Shoenfield's conjecture was refuted (independently) by Lachlan [1966] and C. E. M. Yates [1966], who exhibited a minimal pair in 7—i.e. a pair of elements whose infimum happens to exist and equals the least element. From this point onward, most of the results about 7 stress the pathology of its structure. Received by the editors June 3, 1982. 1980 Mathematics Subject Classification. Primary 03D25, 03D30. ©1984 American Mathematical Society 0025-5726/84 $1.00 + $.25 per page 315 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use