Algebra in superextensions of groups: minimal left ideals

T.O.Banakh; V.Gavrylkiv

We prove that the minimal left ideals of the superextension $\lambda(X)$ of a discrete group $X$ are singletons if and only if $X$ is odd in the sense that each element of $X$ has odd order. On the other hand, the minimal left ideals of the superextension $\lambda(\mathbb Z)$ of the discrete group $\mathbb Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $\lambda(\mathbb Z_2)$ of the compact group $\mathbb Z_2$ of integer 2-adic numbers.

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2. B.~Balcar, F.~Franek, Structural properties of universal minimal dynamical systems for discrete semigroups, Trans. Amer. Math. Soc. 349 (1997) №5, 1697--1724.

3. T.~Banakh, V.~Gavrylkiv, O.~Nykyforchyn, Algebra in superextensions of groups, I: zeros and commutativity, Algebra Discrete Math. (2008), №3, 1--29.

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	Algebra in superextensions of groups: minimal left ideals
Author	T.O.Banakh, V.Gavrylkiv tbanakh@yahoo.com, vgavrylkiv@yahoo.com Ivan Franko National University of Lviv,Universytetska 1, 79000, Ukraine,Vasyl Stefanyk Precarpathian National University,Ivano-Frankivsk, Ukraine
Abstract	We prove that the minimal left ideals of the superextension $\lambda(X)$ of a discrete group $X$ are singletons if and only if $X$ is odd in the sense that each element of $X$ has odd order. On the other hand, the minimal left ideals of the superextension $\lambda(\mathbb Z)$ of the discrete group $\mathbb Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $\lambda(\mathbb Z_2)$ of the compact group $\mathbb Z_2$ of integer 2-adic numbers.
Keywords	algebra, superextension, group, minimal left ideal
DOI	doi:10.30970/ms.31.2.142-148
Reference	1. B.~Balcar, A.~B aszczyk, On minimal dynamical systems on Boolean algebras, Comment. Math. Univ. Carolin. 31 (1990) №1, 7--11. 2. B.~Balcar, F.~Franek, Structural properties of universal minimal dynamical systems for discrete semigroups, Trans. Amer. Math. Soc. 349 (1997) №5, 1697--1724. 3. T.~Banakh, V.~Gavrylkiv, O.~Nykyforchyn, Algebra in superextensions of groups, I: zeros and commutativity, Algebra Discrete Math. (2008), №3, 1--29. 4. T.~Banakh, V.~Gavrylkiv, Algebra in superextension of groups, II: cancelativity and centers, Algebra Discrete Math. (2008), №4. 5. H.~Fustenberg, Y.~Katznelson, Idempotents in compact semigroups and Ramsey Theory, Israel J. Math. 68 (1989), 257--270. 6. V.~Gavrylkiv, The spaces of inclusion hyperspaces over noncompact spaces, Mat. Stud. 28 (2007) №1, 92--110. 7. V.~Gavrylkiv, Right-topological semigroup operations on inclusion hyperspaces, Mat. Stud. 29 (2008) №1, 18--34. 8. N.~Hindman, Finite sums from sequences within cells of partition of $\mathbb N$, J.~Combin. Theory Ser. A 17 (1974) 1--11. 9. N.~Hindman, Ultrafilters and combinatorial number theory, Lecture Notes in Math. 751 (1979) 49--184. 10. N.~Hindman, D.~Strauss, Algebra in the Stone- Cech compactification, de Gruyter, Berlin, New York, 1998. 11. A.~Teleiko, M.~Zarichnyi, Categorical Topology of Compact Hausdofff Spaces, VNTL, Lviv, 1999. 12. I.~Protasov, Combinatorics of Numbers, VNTL, Lviv, 1997. 13. J.~van Mill, Supercompactness and Wallman spaces, Math Centre Tracts. 85. Amsterdam: Math. Centrum., 1977. 14. A.~Verbeek, Superextensions of topological spaces. MC Tract 41, Amsterdam, 1972.
Pages	142-148
Volume	31
Issue	2
Year	2009
Journal	Matematychni Studii
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