Russian Mathematics

, Volume 62, Issue 4, pp 1–12 | Cite as

On Algebras of Distributions of Binary Isolating Formulas for Theories of Abelian Groups and Their Ordered Enrichments

  • K. A. Baikalova
  • D. Yu. Emel’yanov
  • B. Sh. Kulpeshov
  • E. A. Palyutin
  • S. V. Sudoplatov
Article
  • 5 Downloads

Abstract

We describe algebras of distributions of binary isolating formulas for theories of abelian groups and some of their ordered enrichments. The base of this description is the general theory of algebras of isolating formulas. We also take into account the specificity of the basedness of theories of abelian groups on Szmielew invariants. We give Cayley tables for algebras that correspond to theories of basic abelian groups and their ordered enrichments and propose a technique for transforming algebras for theories of basic abelian groups into algebras for arbitrary theories of abelian groups.

Keywords

algebra of distributions of binary isolating formulas abelian group elementary theory ordered enrichment 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Shulepov, I.V. and Sudoplatov, S.V. “Algebras of Distributions for Isolating Formulas of a Complete Theory”, Siberian Elect.Math. Reports 11, 380–407 (2014).MathSciNetMATHGoogle Scholar
  2. 2.
    Sudoplatov, S. V. “Deterministic and Absorbing Algebras”, in 9th Panhellenic Logic Symposium, July 15–18, 2013, National Techn. University of Athens (Greece, Athens, NTUA, 2013), pp. 91–96.Google Scholar
  3. 3.
    Sudoplatov, S.V. Classification of Countable Models of Complete Theories. (NGTU Press, Novosibirsk, 2014), Part 1 [in Russian].MATHGoogle Scholar
  4. 4.
    Ovchinnikova, E. V. and Sudoplatov, S.V. “Generations andQuotients for Algebras of Distributions of Binary Formulas”, Lobachevskii J.Math. 36, No. 4, 403–406 (2015).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Sudoplatov, S. V. “Algebras of Distributions for Binary Semi-Isolating Formulas for Families of Isolated Types and for Countably Categorical Theories”, InternationalMath. Forum 9, No. 21, 1029–1033 (2014).Google Scholar
  6. 6.
    Sudoplatov, S. V. “Forcing of Infinity and Algebras of Distributions of Binary Semi-Isolating Formulas for StronglyMinimal Theories”, Math. and Stat. 2, No. 5, 183–187 (2014).Google Scholar
  7. 7.
    Ovchinnikova, E. V. and Sudoplatov, S. V. “Structures of Distributions of Isolating Formulas as Derivative Structures: for Acyclic Graphs”, in 9th Panhellenic Logic Symposium, July 15–18, 2013, National Techn. University of Athens (Greece, Athens, NTUA, 2013), pp. 74–79.Google Scholar
  8. 8.
    Emel’yanov, D. Yu. “On Algebras of Distributions of Binary Formulas for Theories of Unars”, Izv. Irkutsk. Gos. Un-ta. Ser. Matem. 17, 23–36 (2016).MATHGoogle Scholar
  9. 9.
    Emel’yanov, D. Yu., Kulpeshov, B. Sh., and Sudoplatov, S. V. “Algebras of Distributions for Binary Formulas in Countably CategoricalWeakly o-Minimal Structures”, Algebra Logic 56, No. 1, 13–36 (2017).CrossRefMATHGoogle Scholar
  10. 10.
    Ershov, Yu. L. and Palyutin, E. A. Mathematical Logic (Fizmatlit, Moscow, 2011) [in Russian].MATHGoogle Scholar
  11. 11.
    Kargapolov, M. I. and Merzlyakov, Yu. I. Foundations of Group Theory (Nauka, Moscow, 1982) [in Russian].MATHGoogle Scholar
  12. 12.
    Fuchs, L. Infinite Abelian Groups (Mir, Moscow, 1974), Vol. 1 [Russian. translation].Google Scholar
  13. 13.
    Fuchs, L. Infinite Abelian Groups (Mir, Moscow, 1977), Vol. 2 [Russian. translation].Google Scholar
  14. 14.
    Robinson, A. Complete Theories (North-Holland, Amsterdam, 1956).MATHGoogle Scholar
  15. 15.
    Pillay, A. and Steinhorn, C. “Definable Sets in Ordered Structures”. I, Trans. Amer.Math. Soc. 295, No. 2, 565–592 (1986).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Baldwin, J. T., Blass, A., Glass, A. M. W., and Kueker, D. W. “A ‘Natural’ Theory Without a Prime Model”, Algebra Universalis 3, 152–155 (1973).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Popkov, R. A. “The Distribution of Countable Models of the Theory of the Group of Integers”, Sib. Math. J. 56, No. 1, 155–159 (2015).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  • K. A. Baikalova
    • 1
  • D. Yu. Emel’yanov
    • 2
  • B. Sh. Kulpeshov
    • 3
    • 4
  • E. A. Palyutin
    • 2
    • 4
    • 5
  • S. V. Sudoplatov
    • 1
    • 2
    • 4
    • 5
  1. 1.Novosibirsk State Technical UniversityNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.International Information Technology UniversityAlmatyRepublic of Kazakhstan
  4. 4.Institute of Mathematics and Mathematical ModelingAlmatyRepublic of Kazakhstan
  5. 5.Sobolev Institute of Mathematics SB RASNovosibirskRussia

Personalised recommendations