Degree Spectra of Unary Relations on 〈ω,≤〉

A computable presentation of the linearly ordered set (ω,≤), where ω is the set of natural numbers and ≤ is the natural order on ω, is any linearly ordered set L = (ω,≤L) isomorphic to (ω,≤) such that ≤L is a computable relation. Let X be subset of ω and XL be the image of X in the linear order L under the isomorphism between (ω,≤) and L. The degree spectrum of X is the set of all Turing degrees of XL as one runs over all computable presentations of (ω,≤). In this paper we study the degree spectra of subsets of ω.