Algebra and Logic

, Volume 58, Issue 2, pp 158–172

Degree Spectra of Structures Relative to Equivalences

• P. M. Semukhin
• D. Turetsky
• E. B. Fokina
Article

A standard way to capture the inherent complexity of the isomorphism type of a countable structure is to consider the set of all Turing degrees relative to which the given structure has a computable isomorphic copy. This set is called the degree spectrum of a structure. Similarly, to characterize the complexity of models of a theory, one may examine the set of all degrees relative to which the theory has a computable model. Such a set of degrees is called the degree spectrum of a theory. We generalize these two notions to arbitrary equivalence relations. For a structure $$\mathcal{A}$$ and an equivalence relation E, the degree spectrum DgSp($$\mathcal{A}$$, E) of $$\mathcal{A}$$ relative to E is defined to be the set of all degrees capable of computing a structure $$\mathcal{B}$$ that is E-equivalent to $$\mathcal{A}$$. Then the standard degree spectrum of $$\mathcal{A}$$ is DgSp($$\mathcal{A}$$, ≅) and the degree spectrum of the theory of $$\mathcal{A}$$ is DgSp($$\mathcal{A}$$, ≡). We consider the relations $${\equiv}_{\sum_n}$$ ($$\mathcal{A}{\equiv}_{\sum_n}\mathcal{B}$$ iff the Σn theories of $$\mathcal{A}$$ and $$\mathcal{B}$$ coincide) and study degree spectra with respect to $${\equiv}_{\sum_n}$$.

Keywords

degree spectrum of structure degree spectrum of theory degree spectrum of structure relative to equivalence

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