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A semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup with an identity element is called monoid.

Semigroups are used in various areas of mathematics. $C_0$-semigroups are important in partial differential equations. Semigroups have also connections to automata theory.

Topological (and left/right topological) semigroups are also studied. Perhaps the best know result in this area is Ellis-Numakura lemma. Using Ellis-Numakura lemma, existence of idempotent ultrafilters can be shown.

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