Turing Degrees of Certain Isomorphic Images of Computable Relations

A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A. That is, R is not named in the language of A. We define DgA(R) to be the set of Turing degrees of the images f(R) under all isomorphisms f from A to computable models. We investigate conditions on A and R which are sufficient and necessary for DgA(R) to contain every Turing degree. These conditions imply that if every Turing degree ≤ 000 can be realized in DgA(R) via an isomorphism of the same Turing degree as its image of R, then DgA(R) contains every Turing degree. We also discuss an example of A and R whose DgA(R) coincides with the Turing degrees which are ≤ 00.