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Topologies on $$S$$ determined by idempotents in $$\beta S$$. (English) Zbl 0970.54036
Two topologies are defined and investigated on a semigroup $$S$$ by means of an idempotent free ultrafilter on $$S$$ (idempotent with respect to the operation on $$\beta S$$ extending the operation on $$S$$). One topology is extremally disconneted and both topologies make $$S$$ to a left topological semigroup (in special cases also a right topological semigroup). From other results: If $$S$$ is cancellative then both topologies are Hausdorff. If the defining ultrafilter is strongly right maximal and $$S$$ is a group then both topologies coincide. If the defining ultrafilter is strongly summable and $$S$$ is commutative and cancellative with identity, then the summation is continuous.

##### MSC:
 54H13 Topological fields, rings, etc. (topological aspects) 22A15 Structure of topological semigroups
##### Keywords:
topological semigroup; ultrafilter
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