The smallest ideal K(βS) of the Stone-Cech compactification of a discrete semigroup S is the union of pairwise isomorphic and homeomorphic minimal left ideals. We provide a simple characterization of semigroups for which the smallest ideal of βS is finite and some necessary conditions for the minimal left ideals to be finite. We investigate when the smallest ideal of the Stone-Cech compactification of a Cartesian product can be homeomorphic to a Cartesian product of the smallest ideal of Stone-Cech compactifications. We extend some known results about the fact that, if S is a countably infinite cancellative semigroup, every non-minimal semiprincipal left ideal in βS contains many semiprincipal left ideals defined by right cancelable elements of βS. We conclude with some observations about the topological properties of semiprincipal left ideals in βS.

N/A