Skip to main content
SpringerLink
Log in

Automorphisms of semigroups of k-linked upfamilies

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

A family \( \mathcal{A} \) of non-empty subsets of a set X is called an upfamily, if, for each set \( A\in \mathcal{A} \); any set B ⊃ A belongs to \( \mathcal{A} \). An upfamily \( \mathrm{\mathcal{L}} \) is called k-linked, if \( \cap \mathrm{\mathcal{F}}\ne \varnothing \) for any subfamily \( \mathrm{\mathcal{F}}\subset \mathrm{\mathcal{L}} \) of cardinality \( \left|\mathrm{\mathcal{F}}\right|\le k \). The extension Nk(X) consists of all k-linked upfamilies on X. Any associative binary operation ∗ : X × X → X can be extended to an associative binary operation ∗ : Nk(X) × Nk(X) → Nk(X). Here, we study automorphisms of the extensions of groups and finite monogenic semigroups. We also describe the automorphism groups of extensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Banakh and V. Gavrylkiv, “Algebra in superextension of groups, II: cancelativity and centers,” Algebra Discrete Math., 4, 1–14 (2008).

    MathSciNet  MATH  Google Scholar 

  2. T. Banakh and V. Gavrylkiv, “Algebra in superextension of groups: minimal left ideals,” Mat. Stud., 31, No. 2, 142–148 (2009).

    MathSciNet  MATH  Google Scholar 

  3. T. Banakh and V. Gavrylkiv, “Extending binary operations to functor-spaces,” Carpathian Math. Publ., 1, No. 2, 113–126 (2009).

    MATH  Google Scholar 

  4. T. Banakh and V. Gavrylkiv, “Algebra in the superextensions of twinic groups,” Dissertationes Math. (Rozprawy Mat.), 473, 3–74 (2010).

    MathSciNet  MATH  Google Scholar 

  5. T. Banakh and V. Gavrylkiv, “Algebra in superextensions of semilattices,” Algebra Discrete Math., 13, No. 1, 26–42 (2012).

    MathSciNet  MATH  Google Scholar 

  6. T. Banakh and V. Gavrylkiv, “Algebra in superextensions of inverse semigroups,” Algebra Discrete Math., 13, No. 2, 147–168 (2012).

    MathSciNet  MATH  Google Scholar 

  7. T. Banakh and V. Gavrylkiv, “Characterizing semigroups with commutative superextensions,” Algebra Discrete Math., 17, No. 2, 161–192 (2014).

    MathSciNet  MATH  Google Scholar 

  8. T. Banakh and V. Gavrylkiv, “On structure of the semigroups of k-linked upfamilies on groups,” Asian-Eur. J. Math., 10 (2017), No. 4, https://doi.org/10.1142/S1793557117500838.

  9. T. Banakh and V. Gavrylkiv, “Automorphism groups of superextensions of groups,” Mat. Stud., 48, No. 2 (2017).

  10. T. Banakh, V. Gavrylkiv, and O. Nykyforchyn, “Algebra in superextensions of groups, I: zeros and commutativity,” Algebra Discrete Math., 3, 1–29 (2008).

    MathSciNet  MATH  Google Scholar 

  11. R. Dedekind, “Über Zerlegungen von Zahlen durch ihre grüssten gemeinsammen Teiler,” in: Gesammelte Werke, Vol. 1, Springer, Berlin, 1897, pp. 103–148.

  12. V. Gavrylkiv, “The spaces of inclusion hyperspaces over noncompact spaces,” Mat. Stud., 28, No. 1, 92–110 (2007).

    MathSciNet  MATH  Google Scholar 

  13. V. Gavrylkiv, “Right-topological semigroup operations on inclusion hyperspaces,” Mat. Stud., 29, No. 1, 18–34 (2008).

    MathSciNet  MATH  Google Scholar 

  14. V. Gavrylkiv, “On representation of semigroups of inclusion hyperspaces,” Carpathian Math. Publ., 2, No. 1, 24–34 (2010).

    MathSciNet  MATH  Google Scholar 

  15. V. Gavrylkiv, “Superextensions of cyclic semigroups,” Carpathian Math. Publ., 5, No. 1, 36–43 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Gavrylkiv, “Semigroups of centered upfamilies on finite monogenic semigroups,” J. Algebra, Number Theory: Adv. App., 16, No. 2, 71–84 (2016).

    Article  Google Scholar 

  17. V. Gavrylkiv, “Semigroups of centered upfamilies on groups,” Lobachevskii J. Math., 38, No. 3, 420–428 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Gavrylkiv, “Superextensions of three-element semigroups,” Carpathian Math. Publ., 9, No. 1, 28–36 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. Gavrylkiv, “On the automorphism group of the superextension of a semigroup,” Mat. Stud., 48, No. 1, 3–13 (2017).

    MathSciNet  Google Scholar 

  20. N. Hindman and D. Strauss, Algebra in the Stone– Čech compactification, de Gruyter, Berlin, 1998.

    Book  MATH  Google Scholar 

  21. J. M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995.

    MATH  Google Scholar 

  22. J. van Mill, Supercompactness and Wallman Spaces, Math. Centre, Amsterdam, 1977.

    MATH  Google Scholar 

  23. D. Robinson, A Course in the Theory of Groups, Springer, New York, 1996.

    Book  Google Scholar 

  24. A. Teleiko and M. Zarichnyi, Categorical Topology of Compact Hausdoff Spaces, VNTL, Lviv, 1999.

    MATH  Google Scholar 

  25. A. Verbeek, Superextensions of Topological Spaces, Math. Centre, Amsterdam, 1972.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine

    Volodymyr M. Gavrylkiv

Authors
  1. Volodymyr M. Gavrylkiv
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Volodymyr M. Gavrylkiv.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 14, No. 4, pp. 496–514 October–December, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gavrylkiv, V.M. Automorphisms of semigroups of k-linked upfamilies. J Math Sci 234, 21–34 (2018). https://doi.org/10.1007/s10958-018-3978-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-3978-7

Keywords

Search

Navigation