Ideals and idempotents in the uniform ultrafilters

Abstract

If S is a discrete semigroup, then βS 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 βS, 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.2010 Mathematics Subject Classification. Primary: 54D80, 22A15. Secondary: 03E75, 54H99.