Reductive compactifications of semitopological semigroups (Q1415183)

Let \(S\) be a semitopological semigroup. A pair \((\psi ,X)\) is called a semigroup compactification of \(S\) if \(X\) is a compact Hausdorff right topological semigroup, \(\psi :S\to X\) is a continuous homomorphism with a dense image such that for every \(s\in S\) the mapping \(x\mapsto\psi (s)x\) is continuous. An enveloping semigroup of a semigroup compactification \((\psi ,X)\) determines a semigroup compactification of \(S\) that is isomorphic to \((\psi ,X)\) if and only if \(X\) is right reductive. Any semigroup compactification of \(S\) is right reductive if \(sS\) (or \(Ss\)) is dense in \(S\) for some \(s\in S\). A special semigroup compactification by continuous complex value functions from \(S\) is defined.