Ideals and idempotents in the uniform ultrafilters

Abstract

For every discrete semigroup S, the space βS (the Stone–Čech compactification of S) has a natural, right-topological semigroup structure extending S. Under some mild conditions, U(S), the set of uniform ultrafilters on S, is a two-sided ideal of βS, and therefore contains all of its minimal left ideals and minimal idempotents. Our main theorem states that, if S satisfies some mild distributivity conditions, 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 weakly cancellative and left-cancellative, then S⁎=βS∖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 fairly sharp.