Journal of Symbolic Logic 67 (2):697 - 720 (2002)
We give some new examples of possible degree spectra of invariant relations on Δ 0 2 -categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ 0 2 -categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses  to establish the same result for computable relations on computable linear orderings. We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ 0 2 -categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode  to extend results of Harizanov 
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References found in this work BETA
The D.R.E. Degrees Are Not Dense.S. Barry Cooper, Leo Harrington, Alistair H. Lachlan, Steffen Lempp & Robert I. Soare - 1991 - Annals of Pure and Applied Logic 55 (2):125-151.
Computable Isomorphisms, Degree Spectra of Relations, and Scott Families.Bakhadyr Khoussainov & Richard A. Shore - 1998 - Annals of Pure and Applied Logic 93 (1-3):153-193.
Some Effects of Ash–Nerode and Other Decidability Conditions on Degree Spectra.Valentina S. Harizanov - 1991 - Annals of Pure and Applied Logic 55 (1):51-65.
Turing Degrees of Certain Isomorphic Images of Computable Relations.Valentina S. Harizanov - 1998 - Annals of Pure and Applied Logic 93 (1-3):103-113.
Permitting, Forcing, and Copying of a Given Recursive Relation.C. J. Ash, P. Cholak & J. F. Knight - 1997 - Annals of Pure and Applied Logic 86 (3):219-236.
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