Ideals and idempotents in the uniform ultrafilters

Abstract

If $S$ is a discrete semigroup, then $\beta S$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $\beta S$, and therefore contains all of its minimal left ideals and minimal idempotents. We find some very general conditions under which $U(S)$ contains prime minimal left ideals and left-maximal idempotents. If $S$ is countable, then $U(S) = S^*$, and a special case of our main theorem is that if a countable discrete semigroup $S$ is a weakly cancellative and left-cancellative, then $S^*$ contains prime minimal left ideals and left-maximal idempotents. We will provide examples of weakly cancellative semigroups where these conclusions fail, thus showing that this result is sharp.