Abstract
We construct the degree b ≤ 0″ admitting no algebraic structure with degree spectrum {x: x ≰ b}. Moreover, we solve Miller’s problem of distinguishing incomparable degrees by the spectra of linear orderings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Knight J. F., “Degrees coded in jumps of orderings,” J. Symbolic Logic, 51, No. 4, 1034–1042 (1986).
Richter L. J., “Degrees of structures,” J. Symbolic Logic, 46, No. 4, 723–731 (1981).
Slaman T. A., “Relative to any nonrecursive set,” Proc. Amer. Math. Soc., 126, No. 7, 2117–2122 (1998).
Wehner S., “Enumerations, countable structures, and Turing degrees,” Proc. Amer. Math. Soc., 126, No. 7, 2131–2139 (1998).
Miller R. G., “The Δ 02 -spectrum of a linear order,” J. Symbolic Logic, 66, No. 2, 470–486 (2001).
Downey R., “On presentations of algebraic structures,” in: Complexity, Logic, and Recursion Theory, Dekker, New York, 1997, pp. 157–205.
Soskov I. N., “Degree spectra and co-spectra of structures,” Ann. Univ. Sofia, 96, 45–68 (2003).
Hirschfeldt D. R., Khoussainov B., Shore R. A., and Slinko A. M., “Degree spectra and computable dimensions in algebraic structures,” Ann. Pure Appl. Logic, 115, 71–113 (2002).
Harizanov V. S. and Miller R. G., “Spectra of structures and relations,” J. Symbolic Logic, 72, 324–348 (2007).
Goncharov S. S., Harizanov V. S., Knight J. F., McCoy C., Miller R. G., and Solomon R., “Enumerations in computable structure theory,” Ann. Pure Appl. Logic, 136, 219–246 (2005).
Kalimullin I. Sh., “Spectra of degrees of some algebraic structures,” Algebra and Logic, 46, No. 6, 399–408 (2007).
Soare R. I., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Berlin; Heidelberg; New York; London; Paris; Tokyo (1987).
McEvoy K. and Cooper S. B., “On minimal pairs of enumeration degrees,” J. Symbolic Logic, 50, No. 4, 983–1001 (1985).
Sorbi A., “The enumeration degrees of Σ 02 sets,” in: Complexity, Logic, and Recursion Theory, Dekker, New York, 1997, pp. 303–330.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2008 Kalimullin I. Sh.
The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00605) and the President of the Russian Federation (Grant MK-4314.2008.1).
__________
Kazan’. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 6, pp. 1296–1309, November–December, 2008.
Rights and permissions
About this article
Cite this article
Kalimullin, I.S. Restrictions on the degree spectra of algebraic structures. Sib Math J 49, 1034–1043 (2008). https://doi.org/10.1007/s11202-008-0099-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11202-008-0099-4