Ideals in \(\mathcal{P}_G\) and \(\beta G\) (Q1705843)

Let \(G\) be a discrete group, \(\mathcal{P}_G\) be the Boolean algebra of all subsets of \(G\), \(G^{\ast}=\beta G\backslash G\) is the Stone-Čech compactificaion of \(G\). For an ideal \(\mathcal{I}\) in \(\mathcal{P}_G\), \(\mathcal{I}^{\wedge} = \{p\in \beta G: G\backslash A\in p \text{ for each } A\in \mathcal{I}\}\). Using a classification of the subsets of a group by their size, the authors define in \(\mathcal{P}_G\) the ideals \(Sm_G\), \(Sc_G\), \(Sp_G\) of small, scattered and sparse subsets of \(G\), correspondingly. It is known, that \(Sm_G^{\wedge}=\overline{K(\beta G)}\), \(Sp_G^{\wedge}= \overline{G^{\ast} G^{\ast}}\). \par The authors characterize new ideals in \(\beta G\): 1) between \(\overline{G^{\ast}G^{\ast}}\) and \(G^{\ast}\) and 2) between \(\overline{K(\beta G)}\) and \(\overline{G^{\ast} G^{\ast}}\). For task 1), the authors prove that if a group \(G\) is either countable or Abelian then \(\beta G\) contains no ideals that are maximal among the closed proper subideals of \(G^{\ast}\) (Theorem 3.2), also disproving this statement for the group \(S_{\kappa}\) of all permutations of an infinite cardinal \(\kappa\) (Theorem 3.3). For task 2), the minimal closed ideal \(Sc_G^{\wedge}\) in \(\beta G\) containing all idempotents of \(G^{\ast}\) is characterized (Theorem 4.1).