Ideals in PG and βG

For a discrete group G, we use the natural correspondence between ideals in the Boolean algebra PG of subsets of G and closed subsets in the Stone-Čech compactification βG as a right topological semigroup to introduce and characterize some new ideals in βG. We show that if a group G is either countable or Abelian then βG contains no ideals that are maximal among the closed proper subideals of G⁎, G⁎=βG∖G, but this statement does not hold for the group Sκ of all permutations of an infinite cardinal κ. We characterize the minimal closed ideal in βG containing all idempotents of G⁎.