Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Uniform reducibility of representability problems for algebraic structures

  • 25 Accesses

  • 1 Citations

Abstract

Given a countable algebraic structure \( \mathfrak{B} \) with no degree we find sufficient conditions for the existence of a countable structure \( \mathfrak{A} \) with the following properties: (1) for every isomorphic copy of \( \mathfrak{A} \) there is an isomorphic copy of \( \mathfrak{A} \) Turing reducible to the former; (2) there is no uniform effective procedure for generating a copy of \( \mathfrak{A} \) given a copy of \( \mathfrak{B} \) even having been enriched with an arbitrary finite tuple of constants.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Stukachev A. I., “Degrees of presentability of structures. I,” Algebra and Logic, 46, No. 6, 419–432 (2007).

  2. 2.

    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).

  3. 3.

    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, No. 3, 219–246 (2005).

Download references

Author information

Correspondence to I. Sh. Kalimullin.

Additional information

Original Russian Text Copyright © 2009 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. 50, No. 2, pp. 334–343, March–April, 2009.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kalimullin, I.S. Uniform reducibility of representability problems for algebraic structures. Sib Math J 50, 265–271 (2009). https://doi.org/10.1007/s11202-009-0031-6

Download citation

Keywords

  • computability of an algebraic structure
  • Turing degree
  • mass problem