REGULAR IDEMPOTENTS IN βS

Let S be a discrete semigroup and let beta S be the Stone-Cech compactification of S. We take the points of beta S to be the ultrafilters on S. Being a compact Hausdorff right topological semigroup, beta S has idempotents. Every idempotent p is an element of beta S determines a left invariant topology T(p) on S with a neighborhood base at a is an element of beta S consisting of subsets aB boolean OR {a}, where B is an element of p. If S is a group and p is an idempotent in S* = beta S\S, (S, T(p)) is a homogeneous Hausdorff maximal space. An idernpotent p is an element of beta S is regular if p is uniform and the topology T(p) is regular. We show that for every infinite cancellative semigroup S, there exists a regular idempotent in beta S. As a consequence, we obtain that for every infinite cardinal kappa, there exists a homogeneous regular maximal space of dispersion character kappa. Another consequence says that there exists a translation invariant regular maximal topology on the real line of dispersion character c stronger than the natural topology.