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March 2007 Spectra of structures and relations
Valentina S. Harizanov, Russel G. Miller
J. Symbolic Logic 72(1): 324-348 (March 2007). DOI: 10.2178/jsl/1174668398

Abstract

We consider embeddings of structures which preserve spectra: if g:ℳ →𝒮 with 𝒮 computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on 𝒮). We show that the computable dense linear order ℒ is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph 𝔖. 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 𝔖. Finally, we consider the extent to which all spectra of unary relations on the structure 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 c with c'T 0''.

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Valentina S. Harizanov. Russel G. Miller. "Spectra of structures and relations." J. Symbolic Logic 72 (1) 324 - 348, March 2007. https://doi.org/10.2178/jsl/1174668398

Information

Published: March 2007
First available in Project Euclid: 23 March 2007

zbMATH: 1116.03029
MathSciNet: MR2298485
Digital Object Identifier: 10.2178/jsl/1174668398

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.72 • No. 1 • March 2007
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