Boolean group ideals and the ideal structure of $\beta G$

M.Filali; Ie.Lutsenko; I.V.Protasov

An ideal $\mathcal{I}$ in the Boolean algebra of all subsets of a group $G$ is called a Boolean group ideal if $\mathcal{I}$ contains all finite subsets of $G$ and $A,B\in\mathcal{I}\Rightarrow AB\in\mathcal{I},\ A^{-1}\in\mathcal{I}$. Every Boolean group ideal determines some structure (namely, the group ballean) on $G$ antipodal to the group topology. We show that every countable group admits $2^{2^{\aleph_0}}$ distinct Boolean group ideals, study the lattices of Boolean group ideals and their relationships with T-sequences. The Stone-\v{C}ech compactification $\beta G$ of a discrete group $G$ has a natural structure of a right topological semigroup. We use a duality between the left invariant Boolean ideals on $G$ and the closed left invariant ideals of $\beta G$ to get some information about closed ideals of $\beta G$. In particular, to describe the ideal $\overline{G^*G^*}$ we introduce a new type of subsets of $G$, the sparse subsets.

1. b1 M. Filali, I. Protasov,\/ Ultrafilters and Topologies on Groups, Math. Stud. Monorg. Ser., Vol. 13, VNTL Publishers, Lviv, 2007.

2. b4 R. L. Graham,\/ Rudiments of Ramsey Theory, Regional Conf. Series in Math, Number 45, AMS, Providence, RI, 1981.

3. b5 K. Hart, J. van Mill,\/ Open problems on $\beta\omega$, in Open Problems in Topology, Elsevier, 1990, 98-125.

4. b6 N. Hindman, J. van Mill, P.Simon,\/ Increasing chains of ideals and orbit closures in $\beta\mathbb{Z}$, Proc. Amer. Math. Soc., 114 (1992), 1167-1172.

5. b7 N. Hindman, D. Strauss,\/ Algebra in the Stone-Cech compactification --- Theory and Applications. Walter de Grueter, Berlin, New York, 1998.

6. b14 Ie. Lutsenko, I. V. Protasov,\/ Sparse, thin and other subsets of groups, manuscript.

7. b8 I. Protasov, T. Banakh,\/ Ball Structure and Colorings of Groups and Graphs, Math. Stud. Monorg. Ser., Vol. 11, VNTL Publisher, Lviv, 2003.

8. b9 I. V. Protasov, O. I. Protasova,\/ Sketch of group balleans, Mat. Stud., 22 (2004), 10--20.

9. b10 I. V. Protasov, O. I. Protasova,\/ On closed ideals of $\beta G$, Semigroup Forum, 2007.

10. b11 I. Protasov, M. Zarichniy,\/ General Asymptology, Math. Stud. Monorg. Ser., Vol. 12, VNTL Publisher, Lviv, 2007.

11. b12 I. Protasov, E. Zelenyuk,\/ Topologies on Groups Determined by Sequences, Math. Stud. Monorg. Ser., Vol. 4, VNTL Publisher, Lviv, 1999.

12. b13 I. V. Protasov, E. G. Zelenyuk,\/ Topologies on Abelian groups, Math. USSR Izvestiya, 37 (1991), 445--460. Russian original: Izvestia Akad. Nauk SSSR 54 (1990), 1090--1107.

	Boolean group ideals and the ideal structure of $\beta G$
Author	M.Filali, Ie.Lutsenko, I.V.Protasov Mahmoud.Filali@oulu.fi, protasov@unicyb.kiev.ua Department of Math. Sciences,University of Oulu,P.O. Box 3000, FIN-90014, Finland, Department of Cybernetics,Kyiv University, Volodimirska 64, Kyiv 01033, Ukraine
Abstract	An ideal $\mathcal{I}$ in the Boolean algebra of all subsets of a group $G$ is called a Boolean group ideal if $\mathcal{I}$ contains all finite subsets of $G$ and $A,B\in\mathcal{I}\Rightarrow AB\in\mathcal{I},\ A^{-1}\in\mathcal{I}$. Every Boolean group ideal determines some structure (namely, the group ballean) on $G$ antipodal to the group topology. We show that every countable group admits $2^{2^{\aleph_0}}$ distinct Boolean group ideals, study the lattices of Boolean group ideals and their relationships with T-sequences. The Stone-\v{C}ech compactification $\beta G$ of a discrete group $G$ has a natural structure of a right topological semigroup. We use a duality between the left invariant Boolean ideals on $G$ and the closed left invariant ideals of $\beta G$ to get some information about closed ideals of $\beta G$. In particular, to describe the ideal $\overline{G^G^}$ we introduce a new type of subsets of $G$, the sparse subsets.
Keywords	boolean group ideal, ideal structure, lattice, compactification
DOI	doi:10.30970/ms.31.1.19-28
Reference	1. b1 M. Filali, I. Protasov,\/ Ultrafilters and Topologies on Groups, Math. Stud. Monorg. Ser., Vol. 13, VNTL Publishers, Lviv, 2007. 2. b4 R. L. Graham,\/ Rudiments of Ramsey Theory, Regional Conf. Series in Math, Number 45, AMS, Providence, RI, 1981. 3. b5 K. Hart, J. van Mill,\/ Open problems on $\beta\omega$, in Open Problems in Topology, Elsevier, 1990, 98-125. 4. b6 N. Hindman, J. van Mill, P.Simon,\/ Increasing chains of ideals and orbit closures in $\beta\mathbb{Z}$, Proc. Amer. Math. Soc., 114 (1992), 1167-1172. 5. b7 N. Hindman, D. Strauss,\/ Algebra in the Stone-Cech compactification --- Theory and Applications. Walter de Grueter, Berlin, New York, 1998. 6. b14 Ie. Lutsenko, I. V. Protasov,\/ Sparse, thin and other subsets of groups, manuscript. 7. b8 I. Protasov, T. Banakh,\/ Ball Structure and Colorings of Groups and Graphs, Math. Stud. Monorg. Ser., Vol. 11, VNTL Publisher, Lviv, 2003. 8. b9 I. V. Protasov, O. I. Protasova,\/ Sketch of group balleans, Mat. Stud., 22 (2004), 10--20. 9. b10 I. V. Protasov, O. I. Protasova,\/ On closed ideals of $\beta G$, Semigroup Forum, 2007. 10. b11 I. Protasov, M. Zarichniy,\/ General Asymptology, Math. Stud. Monorg. Ser., Vol. 12, VNTL Publisher, Lviv, 2007. 11. b12 I. Protasov, E. Zelenyuk,\/ Topologies on Groups Determined by Sequences, Math. Stud. Monorg. Ser., Vol. 4, VNTL Publisher, Lviv, 1999. 12. b13 I. V. Protasov, E. G. Zelenyuk,\/ Topologies on Abelian groups, Math. USSR Izvestiya, 37 (1991), 445--460. Russian original: Izvestia Akad. Nauk SSSR 54 (1990), 1090--1107.
Pages	19-28
Volume	31
Issue	1
Year	2009
Journal	Matematychni Studii
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Table of content of issue	html