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

## TL;DR

This paper explores the structure of ideals in the Stone-Čech compactification of a group, revealing differences based on group properties and characterizing minimal closed ideals containing idempotents.

## Contribution

It introduces and characterizes new ideals in ech compactification , especially relating to countable, Abelian, and permutation groups, and analyzes their properties.

## Key findings

- No closed maximal ideals in ech G for countable or Abelian groups.
- Existence of closed maximal ideals in ech G for permutation groups of infinite cardinal.
- Characterization of the minimal closed ideal containing all idempotents.

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