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@ Nik: For any discrete semigroup $(S,.)$, the Stone-Cech compactification $\beta (S)$ of $S$ admits an extension of the operation so that $(\beta S,.)$ is a compact right topological semigroup (meaning right multiplication remains continuous). A good source of information is the book by N.Hindman and D.Struass: books.google.ca/…
Indeed, the proof on MSE is wrong. Theorem 1,/10.2 is about uniqueness, but how about existence? I am trying now to adapt that proof for uniqueness result.
