The scarcity of products in βS ∖ S

Let S be a discrete semigroup and let the Stone-Čech compactification βS of S have the operation extending that of S which makes βS a right topological semigroup with S contained in its topological center. Let S⁎=βS∖S. Algebraically, the set of products S⁎S⁎ tends to be rather large, since it often contains the smallest ideal of βS. We establish here sufficient conditions involving mild cancellation assumptions and assumptions about the cardinality of S for S⁎S⁎ to be topologically small, that is for S⁎S⁎ to be nowhere dense in S⁎, or at least for S⁎∖S⁎S⁎ to be dense in S⁎. And we provide examples showing that these conditions cannot be significantly weakened. These extend results previously known for countable semigroups. Other results deal with large sets missing S⁎S⁎ whose elements have algebraic properties, such as being right cancelable and generating free semigroups in βS.