Algebras of Binary Formulas for Compositions of Theories

We consider algebras of binary formulas for compositions of theories both in the general case and as applied to ℵ0-categorical, strongly minimal, and stable theories, linear preorders, cyclic preorders, and series of finite structures. It is shown that edefinable compositions preserve isomorphisms and elementary equivalence and have basicity formed by basic formulas of the initial theories. We find criteria for e-definable compositions to preserve ℵ0-categoricity, strong minimality, and stability. It is stated that e-definable compositions of theories specify compositions of algebras of binary formulas. A description of forms of these algebras is given relative to compositions with linear orders, cyclic orders, and series of finite structures.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    S. V. Sudoplatov, Classification of Countable Models of Complete Theories, 2nd ed., Novosibirsk State Tech. Univ., Novosibirsk (2018).

    Google Scholar 

  2. 2.

    I. V. Shulepov and S. V. Sudoplatov, “Algebras of distributions for isolating formulas of a complete theory,” Sib. El. Mat. Izv., 11, 380-407 (2014); http://semr.math.nsc.ru/v11/p380-407.pdf.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    S. V. Sudoplatov, “Algebras of distributions for semi-isolating formulas of a complete theory,” Sib. El. Mat. Izv., 11, 408-433 (2014); http://semr.math.nsc.ru/v11/p408-433.pdf.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    S. V. Sudoplatov, “Algebras of distributions for binary semi-isolating formulas for families of isolated types and for countably categorical theories,” Int. Math. Forum, 9, No. 21, 1029-1033 (2014).

    Article  Google Scholar 

  5. 5.

    S. V. Sudoplatov, “Forcing of infinity and algebras of distributions of binary semi-isolating formulas for strongly minimal theories,” Math. Stat., 2, No. 5, 183-187 (2014).

    Google Scholar 

  6. 6.

    B. Sh. Kulpeshov and S. V. Sudoplatov, “Algebras of distributions of binary formulas for quite o-minimal theories,” News Nat. Acad. Sci. Rep. Kazakhstan, Phys.-Math. Ser., 2, No. 300, 5-13 (2015).

    Google Scholar 

  7. 7.

    D. Yu. Emel’yanov, “Algebras of distributions for binary formulas of theories of unars,” Izv. Irkutsk Gos. Univ., Mat., 17, 23-36 (2016).

    MATH  Google Scholar 

  8. 8.

    D. Yu. Emel’yanov, B. Sh. Kulpeshov, and S. V. Sudoplatov, “Algebras of distributions for binary formulas in countably categorical weakly o-minimal structures,” Algebra and Logic, 56, No. 1, 13-36 (2017).

    MathSciNet  Article  Google Scholar 

  9. 9.

    D. Yu. Emel’yanov and S. V. Sudoplatov, “On deterministic and absorbing algebras of binary formulas of polygonometrical theories,” Izv. Irkutsk Gos. Univ., Mat., 20, 32-44 (2017).

    MathSciNet  MATH  Google Scholar 

  10. 10.

    K. A. Baikalova, D. Yu. Emel’yanov, B. Sh. Kulpeshov, E. A. Palyutin, and S. V. Sudoplatov, “On algebras of distributions of binary isolating formulas for theories of Abelian groups and their ordered enrichments,” Izv. Vyssh. Uch. Zav., Mat., No. 4, 3-15 (2018).

    MATH  Google Scholar 

  11. 11.

    D. Yu. Emel’yanov, B. Sh. Kulpeshov, and S. V. Sudoplatov, “Algebras of distributions of binary isolating formulas for quite o-minimal theories,” Algebra and Logic, 57, No. 6, 429-444 (2018).

    MathSciNet  Article  Google Scholar 

  12. 12.

    F. Harary, Graph Theory, Addison-Wesley Ser. Math., Reading, Addison-Wesley, Mass. (1969).

  13. 13.

    S. V. Sudoplatov and E. V. Ovchinnikova, Discrete Mathematics [in Russian], 5th ed., Yurait, Moscow (2020).

    Google Scholar 

  14. 14.

    V. A. Artamonov, V. N. Salii, L. A. Skornyakov, et al., General Algebra [in Russian], Vol. 2, Nauka, Moscow (1991).

    Google Scholar 

  15. 15.

    E. S. Lyapin, Semigroups, Gos. Izd. Fiz.-Mat. Lit., Moscow (1960).

    Google Scholar 

  16. 16.

    S. V. Sudoplatov, “Transitive arrangements of algebraic systems,” Sib. Math. J., 40, No. 6, 1142-1145 (1999).

    MathSciNet  Article  Google Scholar 

  17. 17.

    S. V. Sudoplatov, “Combinations of structures,” Izv. Irkutsk Gos. Univ., Mat., 24, 82-101 (2018).

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Yu. Saffe, E. A. Palyutin, and S. S. Starchenko, “Models of superstable Horn theories,” Algebra and Logic, 24, No. 3, 171-210 (1985).

    MathSciNet  Article  Google Scholar 

  19. 19.

    J. Reineke, “Minimale Gruppen,” Z. Math. Log. Grund. Math., 21, 357-359 (1975).

    MathSciNet  Article  Google Scholar 

  20. 20.

    Y. R. Baisalov, K. A. Meirembekov, and A. T. Nurtazin, “Definably minimal models,” in Model Theory and Algebra, France-Kazakhstan Conf. on Model Theory and Algebra, Astana (2005), pp. 8-11.

  21. 21.

    J. T. Baldwin and A. T. Lachlan, “On strongly minimal sets,” J. Symb. Log., 36, No. 1, 79-96 (1971).

    MathSciNet  Article  Google Scholar 

  22. 22.

    S. Shelah, Classification Theory and the Number of Non-Isomorphic Models, 2nd edn., Stud. Log. Found. Math., 92, North-Holland, Amsterdam (1990).

  23. 23.

    V. Harnik and L. Harrington, “Fundamentals of forking,” Ann. Pure Appl. Log., 26, No. 3, 245-286 (1984).

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to D. Yu. Emel’yanov.

Additional information

D. Yu. Emel’yanov is Supported by RFBR (project No. 20-31-90004), by KN MON RK (grant No. AP08855544), and by SB RAS Fundamental Research Program I.1.1 (project No. 0314-2019-0002).

B. Sh. Kulpeshov is Supported by RFBR (project No. 20-31-90004), by KN MON RK (grant No. AP08855544), and by SB RAS Fundamental Research Program I.1.1 (project No. 0314-2019-0002).

S. V. Sudoplatov is Supported by RFBR (project No. 20-31-90004), by KN MON RK (grant No. AP08855544), and by SB RAS Fundamental Research Program I.1.1 (project No. 0314-2019-0002).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Emel’yanov, D.Y., Kulpeshov, B.S. & Sudoplatov, S.V. Algebras of Binary Formulas for Compositions of Theories. Algebra Logic 59, 295–312 (2020). https://doi.org/10.1007/s10469-020-09602-y

Download citation

Keywords

  • algebra of binary formulas
  • composition of theories
  • e-definable composition
  • 0- categorical theory
  • strongly minimal theory
  • stable theory
  • linear preorder
  • cyclic preorder