We establish that for every computably enumerable (c.e.) Turing degree b, the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings, the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees.
Chubb, Jennifer; Frolov, Andrey; and Harizanov, Valentina, "Degree Spectra of the Successor Relation of Computable Linear Orderings" (2009). Mathematics. 5.