Boolean group ideals and the ideal structure of \(\beta G\). (English) Zbl 1199.22007

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

Mat. Stud. 31, No. 1, 19-28 (2009).

Summary: 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-Čech 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.

MSC:

22A15	Structure of topological semigroups
06E25	Boolean algebras with additional operations (diagonalizable algebras, etc.)

Keywords:

Boolean algebra; Boolean group; Stone-Čech compactification