Distal compactifications of transformation semigroups (Q5939029)

A pair \((S,X)\) is called a transformation semigroup if a binary associative operation on \(S\) is given and a two-sorted binary operation from \(S\times X\) to \(X\) is given such that \(s(tx)=(st)x\) for all \(s,t\in S\) and all \(x\in X\). If \( S\) and \(X\) are topological spaces such that all inner left (or right) translations on \(S\) are continuous and all mappings \((s,-)\) from \(X\) to \(X\) (or \((-,x)\) from \(S\) to \(X\)) are continuous for all \(s\in S\) (or \(x\in X\)) then we say that \((S,X)\) is a left (or right) topological transformation semigroup. If \((S,X)\) is both a left and a right topological transformation semigroup then we say that \((S,X)\) is a semitopological transformation semigroup. A left (or right) topological transformation semigroup \((T,Y)\) is called a left (or right) compactification of a semitopological semigroup \((S,X)\) if \((S,X)\) is a transformation subsemigroup of \((T,Y)\) and \(T\) and/or \(Y\) are compact Hausdorff spaces with a dense subset \(S\) and/or \(X\). A universal right (or left) topological compactification of a semitopological transformation semigroup is described.