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 eS⁎ of ultrafilters on S that converge to e is a semigroup under restriction of the usual operation + on βTd, the Stone-Čech compactification of the discrete semigroup Td. We characterize the smallest ideal of (eS⁎,+), and those sets “central” in (eS⁎,+), that is, those sets which are members of minimal idempotents in (eS⁎,+). We describe some combinatorial applications of those sets that are central in (eS⁎,+).