Ideals and idempotents in the uniform ultrafilters

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.