Foliations induced by congruences (Q1114027)

As the authors point out, every closed right congruence on a Lie group G gives rise to a foliation of G; the leaves of this foliation are the cosets Hg, where H is the equivalence class of the identity. The authors ask whether a similar foliation structure exists for a closed right congruence on an open subsemigroup of the Lie group G. They establish an affirmative result in the case of an open subsemigroup S for which the identity of the group lies in the closure of S.