Semigroup compactification of projective limits of topological semigroups (Q5917681)

For definitions, see \textit{J. H. Carruth, J. A. Hildebrant} and \textit{R. J. Koch} [The theory of topological semigroups. I. Pure and Applied Mathematics, vol. 75 (New York-Basel 1983; Zbl 0515.22003)]. Let \(\{(\eta_\alpha, A_\alpha), \phi^\beta_\alpha\}_{\alpha \leq \beta \in D}\) be a projective system of semigroup compactifications of a topological semigroup \(S\), where \(\eta_\alpha = \phi^\beta_\alpha \eta_\beta\) for every pair \(\alpha \leq \beta\) in a directed set \(D\). Then \(\lim_\to (\eta_\alpha, A_\alpha)\) is a semigroup compactification of \(S\). Furthermore, if \(\{S_\alpha, \phi^\beta_\alpha\}_{\alpha \leq \beta \in D}\) is a projective system of topological semigroups with projective system of semigroup compactifications \(\{(\eta_\alpha, A_\alpha), e^\beta_\alpha\}_{\alpha \leq \beta \in D}\), where \(e^\beta_\alpha \eta_\beta = \eta_\alpha \phi^\beta_\alpha\) for every pair \(\alpha \leq \beta \in D\) such that \(S^* = \lim_\leftarrow S_\alpha\) exists and \(\lambda_\alpha = P_\alpha |_{S^*} :S^*\to S_\alpha\) is surjective for each \(\alpha \in D\), where \(P_\alpha\) is the projection, then \(\lim_\leftarrow (\eta_\alpha, A_\alpha) = A^*\) is a semigroup compactification of \(S^*\).