The semigroup of ultrafilters near an idempotent of a semitopological semigroup

Let ( T , + ) be a Hausdorff semitopological semigroup, S be a dense subsemigroup of T and e be an idempotent element of T. The set e S ⁎ of ultrafilters on S that converge to e is a semigroup under restriction of the usual operation + on β T d , the Stone–Čech compactification of the discrete semigroup T d . We characterize the smallest ideal of ( e S ⁎ , + ) , and those sets “central” in ( e S ⁎ , + ) , that is, those sets which are members of minimal idempotents in ( e S ⁎ , + ) . We describe some combinatorial applications of those sets that are central in ( e S ⁎ , + ) .