Embeddings of [formula omitted] and the contiguous degrees

Abstract

Downey and Lempp (J. Symbolic Logic 62 (1997) 1215–1240) have shown that the contiguous computably enumerable (c.e.) degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truth-table degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5-element lattice N 5 into the c.e. degrees which maps the top to the degree a. In particular, this shows that local nondistributivity coincides with local nonmodularity in the computably enumerable degrees.