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Spectra of structures and relations. (English) Zbl 1116.03029
Summary: We consider embeddings of structures which preserve spectra: if \(g:{\mathcal M}\to{\mathcal S}\) with \({\mathcal S}\) computable, then \({\mathcal M}\) should have the same Turing degree spectrum (as a structure) that \(g({\mathcal M})\) has (as a relation on \({\mathcal S})\). We show that the computable dense linear order \({\mathcal L}\) is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph \({\mathcal G}\). Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on \({\mathcal G}\). Finally, we consider the extent to which all spectra of unary relations on the structure \({\mathcal L}\) may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees \({\mathbf c}\) with \({\mathbf c}'\geq_T{\mathbf 0}''\).

MSC:
03C57 Computable structure theory, computable model theory
03D28 Other Turing degree structures
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