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Degree spectra of intrinsically c.e. relations

Published online by Cambridge University Press:  12 March 2014

Denis R. Hirschfeldt
Denis R. Hirschfeldt*
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
*
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA, E-mail: [email protected]
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Abstract

We show that for every c.e. degree a > 0 there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. This result can be extended in two directions. First we show that for every uniformly c.e. collection of sets S there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is the set of degrees of elements of S. Then we show that if α ∈ ω ∪ {ω} then for any α-c.e. degree a > 0 there exists an intrinsically α-c.e. relation on the domain of a computable structure whose degree spectrum {0, a}. All of these results also hold for m-degree spectra of relations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 441 - 469
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Ash, C. J., Cholak, P., and Knight, J. F., Permitting, forcing, and copying of a given recursive relation, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 219–236.CrossRefGoogle Scholar
[2]Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Clayton, 1979) (Crossley, J. N., editor), Upside Down A Book Company, Yarra Glen, Australia, 1981, pp. 26–41.Google Scholar
[3]Barker, E., Intrinsically relations, Annals of Pure and Applied Logic, vol. 39 (1988), pp. 105–130.CrossRefGoogle Scholar
[4]Cholak, P., Goncharov, S. S., Khoussainov, B., and Shore, R. A., Computably categorical structures and expansions by constants, this Journal, vol. 64 (1999), pp. 13–37.Google Scholar
[5]Cooper, S. B., Harrington, L., Lachlan, A. H., Lempp, S., and Soare, R. I., The d.r.e. degrees are not dense, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 125–151.CrossRefGoogle Scholar
[6]Epstein, R. L., Haas, R., and Kramer, R. L., Hierarchies of sets and degrees below 0′, Logic year 1979–80 (Proceedings, Seminars and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80) (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, no. 859, Springer-Verag, Heidelberg, 1981, pp. 32–48.CrossRefGoogle Scholar
[7]Ershov, Yu. L., Goncharov, S. S., Nerode, A., and Remmel, J.B. (editors), Handbook of recursive mathematics, Studies in Logic and Foundational Mathematics, Elsevier Science, Amsterdam, 1998.Google Scholar
[8]Goncharov, S. S., Autostability of models and abelian groups, Algebra and Logic, vol. 19 (1980), pp. 13–27.CrossRefGoogle Scholar
[9]Goncharov, S. S., Computable single-valued numerations, Algebra and Logic, vol. 19 (1980), pp. 325–356.CrossRefGoogle Scholar
[10]Goncharov, S. S., Problem of the number of non-self-equivalent constructivizations, Algebra and Logic, vol. 19 (1980), pp. 401–414.CrossRefGoogle Scholar
[11]Goncharov, S. S. and Khoussainov, B., On the spectrum of degrees of decidable relations, Doklady mathematics, vol. 55 (1997), pp. 55–57, research announcement.Google Scholar
[12]Harizanov, V. S., The possible turing degree of the nonzero member in a two element degree spectrum, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 1–30.CrossRefGoogle Scholar
[13]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimension in algebraic structures, to appear.Google 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]Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar