Generalized nonsplitting in the recursively enumerable degrees

Abstract

AbstractWe investigate the algebraic structure of the upper semi-lattice formed by the recursively enumerable Turing degrees. The following strong generalization of Lachlan's Nonsplitting Theorem is proved: Given n ≥ 1, there exists an r.e. degree d such that the interval [d, 0′] ⊂ R admits an embedding of the n-atom Boolean algebra preserving (least and) greatest element, but also such that there is no (n + 1 )-tuple of pairwise incomparable r.e. degrees above d which pairwise join to 0′ (and hence, the interval [d, 0′] ⊂ R does not admit a greatest-element-preserving embedding of any lattice which has n + 1 co-atoms, including ). This theorem is the dual of a theorem of Ambos-Spies and Soare, and yields an alternative proof of their result that the theory of R has infinitely many one-types.