Ideals in $\mathcal{P} _G$ and $βG$

Igor Protasov, Ksenia Protasova

Published 2017-04-08Version 1

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 $G^*$, $G^* = \beta G \setminus G$, 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 $G^*$.