The cardinality of \(\beta_{A}(S_{J})\) (Q1935463)

Let \(J\) be an infinite set, let \(\mathcal P_f(J)\) denote the collection of all nonempty finite subsets of~\(J\), and let \(S_J=\{(i,f):i\in\mathcal P_f(J)\) and \(f:\mathcal P(i)\to \mathcal P(i)\}\). Let \(\ast\) be the operation on~\(S_J\) defined by \((i,f)\ast(k,g)=(i\cup k,f\ast g)\) where \(f\ast g:\mathcal P(i\cup k)\to\mathcal P(i\cup k)\) is defined by \((f\ast g)(x)=g(x\cap k)\), if \(x=\emptyset\) or \(x\cap k\neq\emptyset\), and \((f\ast g)(x)=f(x)\), otherwise. The author considers the Stone-Čech compactification \((\beta S_J,{\circledast})\) of the discrete right topological semigroup \((S_J,{\ast})\) and proves that there is an injective map \(A\mapsto\beta_A(S_J)\) of \(\mathcal P(J)\to\mathcal P(\beta S_J)\) such that each \(\beta_A(S_J)\) is a~closed subsemigroup of \((\beta S_J,{\circledast})\), \(\beta_A(S_J)\) is the smallest ideal of \((\beta S_J,{\circledast})\) in the case when \(A=J\), \(\{\beta_A(S_J):A\in\mathcal P(J)\}\) is a~partition of \(\beta S_J\), and \(|\beta_A(S_J)|=2^{2^{|J|}}\) for all \(A\subseteq J\). These structural properties of \((\beta S_J,{\circledast})\) are similar to the already known properties of the Stone-Čech compactification \((\beta S_J,{\biguplus})\) of the discrete semigroup \((I,{\cup})\).