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Spectra of structures and relations

Published online by Cambridge University Press:  12 March 2014

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
Corresponding

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
Copyright
Copyright © Association for Symbolic Logic 2007

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References

[1]Ash, C. J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.Google Scholar
[2]Chisholm, J., The complexity of intrinsically r.e. subsets of existentially decidable models, this Journal, vol. 55 (1990), pp. 1213–1232.Google Scholar
[3]Csima, B. F., Harizanov, V. S., Miller, R. G., and Montalbán, A., Computability of Fraïssé limits, to appear.Google Scholar
[4]Downey, R. G., Goncharov, S. S., and Hirschfeldt, D. R., Degree spectra for relations on Boolean algebras, Algebra and Logic, vol. 42 (2003), pp. 105–111.CrossRefGoogle Scholar
[5]Downey, R. G. and Jockusch, C. G., Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871–880.CrossRefGoogle Scholar
[6]Downey, R. G. and Knight, J. F., Orderings with α-th jump degree 0(α), Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 545–552.Google Scholar
[7]Goncharov, S. S., Harizanov, V. S., Knight, J. F., McCoy, C., Miller, R. G., and Solomon, R., Enumerations in computable structure theory, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 219–246.CrossRefGoogle Scholar
[8]Harizanov, V. S., Uncountable degree spectra, Annals of Pure Applied Logic, vol. 54 (1991), pp. 255–263.CrossRefGoogle Scholar
[9]Harizanov, V. S., Pure computable model theory, Handbook of recursive mathematics, vol. 1, Studies in Logic and the Foundations of Mathematics, vol. 138, Elsevier, Amsterdam, 1998, pp. 3–114.Google Scholar
[10]Harizanov, V. S., Relations on computable structures, Contemporary mathematics (Bokan, N., editor), University of Belgrade, 2000, pp. 65–81.Google Scholar
[11]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., And Slinko, A. M., Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.Google Scholar
[12]Hodges, W., A shorter model theory, Cambridge University Press, Cambridge, 1997.Google Scholar
[13]Jockusch, C. G. Jr. and Soare, R. I., Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.CrossRefGoogle Scholar
[14]Khoussainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations, and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153–193.CrossRefGoogle Scholar
[15]Khoussainov, B. and Shore, R. A., Effective model theory: the number of models and their complexity, Models and computability: Invited papers from logic colloquium '97 (Cooper, S.B. and Truss, J.K., editors), London Mathematical Society Lecture Note Series, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193–240.Google Scholar
[16]Knight, J. F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), pp. 1034–1042.Google Scholar
[17]Miller, R., The -spectrum of a linear order, this Journal, vol. 66 (2001), pp. 470–486.Google Scholar
[18]Moses, M., Relations intrinsically recursive in linear orders, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 467–472.CrossRefGoogle Scholar
[19]Richter, L. J., Degrees of structures, this Journal, vol. 46 (1981), pp. 723–731.Google Scholar
[20]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar

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