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 ⁎ = β S ∖ 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.