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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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##### References:
 [1] Recursively enumerable sets and degrees (1987) [2] Contemporary mathematics pp 65–81– (2000) [3] Annals of Pure Applied Logic 54 pp 255–263– (1991) [4] Annals of Pure and Applied Logic 136 pp 219–246– (2005) [5] Proceedings of the American Mathematical Society 114 pp 545–552– (1992) [6] Proceedings of the American Mathematical Society 122 pp 871–880– (1994) · Zbl 0805.90016 [7] Algebra and Logic 42 pp 105–111– (2003) [8] The complexity of intrinsically r.e. subsets of existentially decidable models 55 pp 1213–1232– (1990) [9] Computable structures and the hyperarithmetical hierarchy (2000) · Zbl 0960.03001 [10] Degrees of structures 46 pp 723–731– (1981) [11] Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32 pp 467–472– (1986) [12] The -spectrum of a linear order 66 pp 470–486– (2001) [13] Degrees coded in jumps of orderings 51 pp 1034–1042– (1986) [14] Models and computability: Invited papers from logic colloquium ’97 259 pp 193–240– (1999) [15] Annals of Pure and Applied Logic 93 pp 153–193– (1998) [16] Annals of Pure and Applied Logic 52 pp 39–64– (1991) [17] A shorter model theory (1997) · Zbl 0873.03036 [18] Annals of Pure and Applied Logic 115 pp 71–113– (2002) [19] Handbook of recursive mathematics, vol. 1 138 pp 3–114– (1998) · Zbl 0930.03037
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