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Degrees of bi-embeddable categoricity of equivalence structures

  • Nikolay Bazhenov
  • Ekaterina Fokina
  • Dino RosseggerEmail author
  • Luca San Mauro
Open Access
Article

Abstract

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) \(\Delta ^0_\alpha \) bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of \(\Delta ^0_\alpha \) bi-embeddable categoricity and relative \(\Delta ^0_\alpha \) bi-embeddable categoricity coincide for equivalence structures for \(\alpha =1,2,3\). We also prove that computable equivalence structures have degree of bi-embeddable categoricity \(\mathbf {0},\mathbf {0}'\), or \(\mathbf {0}''\). We furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.

Keywords

Computable categoricity Bi-embeddability Degrees of categoricity Degrees of bi-embeddable categoricity 

Mathematics Subject Classification

03C57 

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). The first author was supported by the Russian Foundation for Basic Research, according to the research Project No. 16-31-60058 mol_a_dk. The second, third and fourth author were supported by the Austrian Science Fund FWF through project P 27527.

References

  1. 1.
    Fröhlich, A., Shepherdson, J.C.: Effective procedures in field theory. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 248(950), 407–432 (1956)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Maltsev, A.I.: On recursive Abelian groups. In: Soviet Mathematics, vol. 3. Doklady (1962)Google Scholar
  3. 3.
    Fokina, E., Harizanov, V., Melnikov, A.G.: Computable model theory. Turing’s legacy: developments from Turing’s ideas in logic 42, 124–191 (2014)Google Scholar
  4. 4.
    Montalbán, A.: Up to equimorphism, hyperarithmetic is recursive. J. Symb. Log. 70, 360–378 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Montalbán, A.: On the equimorphism types of linear orderings. Bull. Symb. Log. 13, 71–99 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Greenberg, N., Montalbán, A.: Ranked structures and arithmetic transfinite recursion. Trans. Am. Math. Soc. 360(3), 1265–1307 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fokina, E., Rossegger, D., Mauro, L.S.: Bi-embeddability spectra and bases of spectra. arXiv preprint arXiv:1808.05451 (2018)
  8. 8.
    Robert, I.S.: Turing Computability. Theory and Applications of Computability. Springer, Berlin Heidelberg (2016)zbMATHGoogle Scholar
  9. 9.
    Calvert, W., Cenzer, D., Harizanov, V., Morozov, A.: Effective categoricity of equivalence structures. Ann. Pure Appl. Log. 141, 61–78 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fokina, E., Kalimullin, I.S., Miller, R.: Degrees of categoricity of computable structures. Arch. Math. Log. 49(1), 51–67 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Russell, M.: D-computable categoricity for algebraic fields. J. Symb. Log. 74(12), 1325–1351 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fokina, E., Frolov, A., Kalimullin, I.S.: Categoricity spectra for rigid structures. Notre Dame J. Form. Log. 57(1), 45–57 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Csima, B.F., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy. Notre Dame J. Form. Log. 54(2), 215–231 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Csima, B.F., Stephenson, J.: Finite computable dimension and degrees of categoricity. Annals of Pure and Applied Logic (2018)Google Scholar
  15. 15.
    Bazhenov, N.A., Kalimullin, I.S., Yamaleev, M.M.: Degrees of categoricity vs. strong degrees of categoricity. Algebra Log. 55(2), 173–177 (2016)CrossRefGoogle Scholar
  16. 16.
    Bazhenov, N.A., Kalimullin, I.S., Yamaleev, M.M.: Degrees of categoricity and spectral dimension. J. Symb. Log. 83(1), 103–116 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kach, A.M., Turetsky, D.: \({\Delta ^{0}}_2\)-categoricity of equivalence structures. N. Z. J. Math. 39, 143–149 (2009)zbMATHGoogle Scholar
  18. 18.
    Downey, R.G., Melnikov, A.G., Meng Ng, K.: On \({\Delta ^{0}}_2\)-categoricity of equivalence relations. Ann. Pure Appl. Log. 166(9), 851–880 (2015)CrossRefGoogle Scholar
  19. 19.
    Rodney, G.D., Asher, M.K., Turetsky, D.: Limitwise monotonic functions and their applications. In: Proceedings Of The 11Th Asian Logic Conference, pp. 59–85 (2011)Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Institute of Discrete Mathematics and GeometryVienna University of TechnologyWienAustria

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