The Quotient Semilattice of the Recursively Enumerable Degrees Modulo the Cappable Degrees

Steven Schwarz
1984 Transactions of the American Mathematical Society  
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
more » ... 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
doi:10.2307/2000006 fatcat:ms3vfscxfnb5bh3jzrxhbojvrm