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Spectra of structures and relations

Published online by Cambridge University Press:  12 March 2014

Valentina S. Harizanov  and
Russell G. Miller
Valentina S. Harizanov
Affiliation:
Department of Mathematics, The George Washington University, Washington, DC 20052, USA. E-mail: harizanv@gwu.edu Department of Mathematics, Queens College– C.U.N.Y., 65-30 Kissena Blvd. Flushing, New York 11367, USA
Russell G. Miller
Affiliation:
PH.D. Program in Computer Science, The Graduate Center of C.U.N.Y., 365 Fifth Avenue, New York, New York 10016, USA. E-mail: Russell.Miller@qc.cuny.edu
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Abstract

We consider embeddings of structures which preserve spectra: if g : ℳ → with computable, then ℳ should have the same Turing degree spectrum (as a structure) that g(ℳ) has (as a relation on ). 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 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 . 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′ ≥ τ 0″.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 324 - 348
Copyright
Copyright © Association for Symbolic Logic 2007

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