The center and extended center of the maximal groups in the smallest ideal of \(\beta\mathbb N\) (Q2850644)

The addition on the set \(\mathbb N\) of positive integers extends to \(\beta \mathbb N\) and the operation \(\rho_p: \beta \mathbb N\to \beta \mathbb N\) defined by \(\rho_p(q)=p+q\) for any \(q\in \beta \mathbb N\), turns out to be continuous, i.e., \((\beta \mathbb N,+)\) is a right topological semigroup. The authors study the set \(K(\beta \mathbb N)=\bigcup\{L: L\) is a minimal left ideal of \(\beta \mathbb N\}\); it is known that also \(K(\beta \mathbb N)=\bigcup\{L: L\) is a minimal right ideal of \(\beta \mathbb N\}\). Given a set \(A\subseteq \beta \mathbb N\), let \(E(A)=\{q\in A: q+q=q\}\). The paper's purpose is to study the set \(E(K(\beta \mathbb N))\).NEWLINENEWLINEIf \(q\in E(K(\beta \mathbb N))\), then \(G_q=q+\beta \mathbb N+q\) and \(D_q=\{u\in \mathbb N^*: u+v=v+u\) for any \(v\in G_q\}\). Furthermore, \(I=\bigcap_{n=1}^\infty \overline{n\cdot \mathbb N}\). The authors prove, among other things, that if, for all elements \(q\in E(K(\beta \mathbb N))\), we have the inclusion \(D_q\cap I \subseteq E(\beta \mathbb N)\), then there is no nontrivial continuous homomorphism from \(\beta \mathbb N\) to \(\mathbb N^*\).