Ideals and idempotents in the uniform ultrafilters

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 ide...

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Bibliographic Details
Published in:Topology and its applications 2018-03, Vol.237, p.53-66
Main Author: Brian, William R.
Format: Article
Language:English
Subjects:
Left-maximal idempotents
Minimal left ideals
Stone–Čech compactification
Uniform ultrafilters
Weak [formula omitted]-sets
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Summary: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.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2018.01.012