Ideals in ${\mathcal{P}}_G$ and $\beta G$

Abstract

For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra ${\mathcal{P}}_G$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $\beta G$ as a right topological semigroup to introduce and characterize some new ideals in $\beta G$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $\beta G$ maximal in $G^{\ast }$, $G^{\ast }=\beta G\setminus G$, but this statement does not hold for the group $S_{\kappa }$ of all permutations of an infinite cardinal $\kappa$. We characterize the minimal closed ideal in $\beta G$ containing all idempotents of $G^{\ast }$.

Keywords: Stone-$\check{C}$ech compactification, Boolean algebra, ideal, filter, ultrafilter