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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
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