On topological congruences of a topological semigroup (Q411968)

It is a famous problem: ``When is the quotient semigroup \(\frac{S}{\theta}\) with the quotient topology a topological semigroup?''. Here, \(S\) is a Hausdorff topological semigroup and \(\theta\) is a closed congruence on \(S\). This problem was studied by Wallace, and later by Lawson and Madison. Almost all the works about this question used some strong topological assumptions and they rarely used the algebraic structures of topological semigroups. However, the author claims that he considers both of these structures for a complete answer. He investigates conditions which are closely related to the ideals and topological structures of \(S\). First, he gives a necessary and sufficient condition on \(S\) and a closed ideal \(I\) of \(S\) such that \(\frac{S}{I}\) is a \(\kappa_\omega\)-space, then he uses this result to generalize Lawson and Madison's well known theorem.