On Lachlan's major sub-degree problem

The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become the longest-standing open question concerning the structure of the computably enumerable (c.e.) degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, 0′ = b ∨ x if and only if 0′ = a ∨ x. In this paper, we show that every c.e. degree b 6= 0 or 0′ has a major subdegree, answering Lachlan’s question affirmatively.