On the automorphism group of the superextension of a semigroup

V. M. Gavrylkiv

doi:10.15330/ms.48.1.3-13

A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an {\it upfamily} if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be {\it linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is {\it maximal linked} if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The {\it superextension} $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* \colon X\times X \to X$ can be extended to an associative binary operation $*\colon \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study automorphisms of superextensions of semigroups and describe the automorphism groups of superextensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups. Also we find the automorphism groups of superextensions of all semigroups $S$ of order $|S|\leq3$.

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

2. T. Banakh, V. Gavrylkiv, Algebra in superextension of groups: minimal left ideals, Mat. Stud. 31 (2009), 142-148.

3. T. Banakh, V. Gavrylkiv, Algebra in the superextensions of twinic groups, Dissertationes Math., 473 (2010), 3-74.

4. T. Banakh, V. Gavrylkiv, Algebra in superextensions of semilattices, Algebra Discrete Math., 13 (2012), №1, 26-42.

5. T. Banakh, V. Gavrylkiv, Algebra in superextensions of inverse semigroups, Algebra Discrete Math., 13 (2012), №2, 147-168.

6. T. Banakh, V. Gavrylkiv, Characterizing semigroups with commutative superextensions, Algebra Discrete Math., 17 (2014), №2, 161-192.

7. T. Banakh, V. Gavrylkiv, On structure of the semigroups of k-linked upfamilies on groups, Asian- European J. Math., 10 (2017), №2, 1750083, 15 p.

8. T. Banakh, V. Gavrylkiv, Automorphism groups of superextensions of groups, preprint.

9. T. Banakh, V. Gavrylkiv, Automorphism groups of superextensions of finite monogenic semigroups, preprint.

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

11. A.H. Clifford, G.B. Preston, The algebraic theory of semigroups, V.I, Mathematical Surveys, V.7, (AMS, Providence, RI, 1961).

12. R. Dedekind, Uber Zerlegungen von Zahlen durch ihre grЃNussten gemeinsammen Teiler, In Gesammelte Werke, Bd. 1 (1897), 103-148.

13. F. Diego, K.H. Jonsdottir, Associative operations on a three-element set, TMME, 5 (2008), №2&3, 257-268.

14. V. Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces, Mat. Stud., 28 (2007), №1, 92-110.

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

16. V. Gavrylkiv, Monotone families on cyclic semigroups, Precarpathian Bull. Shevchenko Sci. Soc., 17 (2012), №1, 35-45.

17. V. Gavrylkiv, Superextensions of cyclic semigroups, Carpathian Math. Publ., 5 (2013), №1, 36-43.

18. V. Gavrylkiv, Semigroups of linked upfamilies, Precarpathian Bull. Shevchenko Sci. Soc., 29 (2015), №1, 104.112.

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

20. V. Gavrylkiv, Semigroups of centered upfamilies on groups, Lobachevskii J. Math., 38 (2017), №3, 420-428.

21. V. Gavrylkiv, Superextensions of three-element semigroups, Carpathiam Math. Publ., 9 (2017), №1, 28-36.

22. N. Hindman, D. Strauss, Algebra in the Stone-.Cech compactification, (de Gruyter, Berlin, New York, 1998).

23. J.M. Howie, Fundamentals of semigroup theory, (The Clarendon Press, Oxford University Press, New York, 1995).

24. J. van Mill, Supercompactness and Wallman spaces, V.85, of Math. Centre Tracts, Math. Centrum, Amsterdam, 1977.

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

26. A. Verbeek, Superextensions of topological spaces, V.41 of Math. Centre Tracts, Math. Centrum, Amsterdam, 1972.

	On the automorphism group of the superextension of a semigroup
Author	V. M. Gavrylkiv vgavrylkiv@gmail.com Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Abstract	A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an {\it upfamily} if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be {\it linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is {\it maximal linked} if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The {\it superextension} $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* \colon X\times X \to X$ can be extended to an associative binary operation $*\colon \lambda(X)\times\lambda(X)\to\lambda(X)$. In the paper we study automorphisms of superextensions of semigroups and describe the automorphism groups of superextensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups. Also we find the automorphism groups of superextensions of all semigroups $S$ of order $\|S\|\leq3$.
Keywords	semigroup; maximal linked upfamily; superextension; automorphism group
DOI	doi:10.15330/ms.48.1.3-13
Reference	1. T. Banakh, V. Gavrylkiv, Algebra in superextension of groups, II: cancelativity and centers, Algebra Discrete Math., 4 (2008), 1-14. 2. T. Banakh, V. Gavrylkiv, Algebra in superextension of groups: minimal left ideals, Mat. Stud. 31 (2009), 142-148. 3. T. Banakh, V. Gavrylkiv, Algebra in the superextensions of twinic groups, Dissertationes Math., 473 (2010), 3-74. 4. T. Banakh, V. Gavrylkiv, Algebra in superextensions of semilattices, Algebra Discrete Math., 13 (2012), №1, 26-42. 5. T. Banakh, V. Gavrylkiv, Algebra in superextensions of inverse semigroups, Algebra Discrete Math., 13 (2012), №2, 147-168. 6. T. Banakh, V. Gavrylkiv, Characterizing semigroups with commutative superextensions, Algebra Discrete Math., 17 (2014), №2, 161-192. 7. T. Banakh, V. Gavrylkiv, On structure of the semigroups of k-linked upfamilies on groups, Asian- European J. Math., 10 (2017), №2, 1750083, 15 p. 8. T. Banakh, V. Gavrylkiv, Automorphism groups of superextensions of groups, preprint. 9. T. Banakh, V. Gavrylkiv, Automorphism groups of superextensions of finite monogenic semigroups, preprint. 10. T. Banakh, V. Gavrylkiv, O. Nykyforchyn, Algebra in superextensions of groups, I: zeros and commutativity, Algebra Discrete Math., 3 (2008), 1-29. 11. A.H. Clifford, G.B. Preston, The algebraic theory of semigroups, V.I, Mathematical Surveys, V.7, (AMS, Providence, RI, 1961). 12. R. Dedekind, Uber Zerlegungen von Zahlen durch ihre grЃNussten gemeinsammen Teiler, In Gesammelte Werke, Bd. 1 (1897), 103-148. 13. F. Diego, K.H. Jonsdottir, Associative operations on a three-element set, TMME, 5 (2008), №2&3, 257-268. 14. V. Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces, Mat. Stud., 28 (2007), №1, 92-110. 15. V. Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces, Mat. Stud., 29 (2008), №1, 18-34. 16. V. Gavrylkiv, Monotone families on cyclic semigroups, Precarpathian Bull. Shevchenko Sci. Soc., 17 (2012), №1, 35-45. 17. V. Gavrylkiv, Superextensions of cyclic semigroups, Carpathian Math. Publ., 5 (2013), №1, 36-43. 18. V. Gavrylkiv, Semigroups of linked upfamilies, Precarpathian Bull. Shevchenko Sci. Soc., 29 (2015), №1, 104.112. 19. V. Gavrylkiv, Semigroups of centered upfamilies on finite monogenic semigroups, J. Algebra, Number Theory: Adv. App., 16 (2016), №2, 71-84. 20. V. Gavrylkiv, Semigroups of centered upfamilies on groups, Lobachevskii J. Math., 38 (2017), №3, 420-428. 21. V. Gavrylkiv, Superextensions of three-element semigroups, Carpathiam Math. Publ., 9 (2017), №1, 28-36. 22. N. Hindman, D. Strauss, Algebra in the Stone-.Cech compactification, (de Gruyter, Berlin, New York, 1998). 23. J.M. Howie, Fundamentals of semigroup theory, (The Clarendon Press, Oxford University Press, New York, 1995). 24. J. van Mill, Supercompactness and Wallman spaces, V.85, of Math. Centre Tracts, Math. Centrum, Amsterdam, 1977. 25. A. Teleiko, M. Zarichnyi, Categorical Topology of Compact Hausdoff Spaces, VNTL, Lviv, 1999. 26. A. Verbeek, Superextensions of topological spaces, V.41 of Math. Centre Tracts, Math. Centrum, Amsterdam, 1972.
Pages	3-13
Volume	48
Issue	1
Year	2017
Journal	Matematychni Studii
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Table of content of issue	html