Embedding locally compact semigroups into groups (Q1266919)

A semigroup (or a group) \(S\) endowed with a topology \(\tau\) is a semitopological semigroup (or a semitopological group) if all left and right inner translations of \(S\) are continuous, and \(S\) is a topological semigroup (or a topological group, respectively) if the binary operation is a continuous mapping from \(S^2\) with the product topology into \(S\). Assume that \(S\) is a semigroup and \(G\) is a group such that \(S\) is a subsemigroup of \(G\) generating \(G\). If \(S\) is a semitopological semigroup then there exists a topology \(\tau\) on \(G\) such that \(G\) is a semitopological group and \(S\) is an open subspace of \(G\) if and only if all left and right inner translations of \(S\) are open. In this case, \(G\) is Hausdorff just when \(S\) is Hausdorff, \(G\) is a topological group just when \(S\) is a topological semigroup and if \(S\) is locally compact and Hausdorff then \(G\) is a locally compact Hausdorff topological group. Let \(I_0\) be the union of all open subspaces \(U\) of \(S\) such that \(sU\), \(Ut\) and \(sUt\) are open for all \(s,t\in S\). If \(I_0\neq\emptyset\) then \(G\) admits a unique topology \(\tau\) such that \(G\) is a locally compact topological group, \(I_0\) is an open subspace of \(G\) and the inclusion from \(S\) into \(G\) is continuous.