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Monoid
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== Acts and operator monoids == {{main|monoid act}} Let {{math|''M''}} be a monoid, with the binary operation denoted by {{math|β’}} and the identity element denoted by {{math|''e''}}. Then a (left) '''{{math|''M''}}-act''' (or left act over {{math|''M''}}) is a set {{math|''X''}} together with an operation {{math|β : ''M'' Γ ''X'' β ''X''}} which is compatible with the monoid structure as follows: * for all {{math|''x''}} in {{math|''X''}}: {{math|1=''e'' β ''x'' = ''x''}}; * for all {{math|''a''}}, {{math|''b''}} in {{math|''M''}} and {{math|''x''}} in {{math|''X''}}: {{math|1=''a'' β (''b'' β ''x'') = (''a'' β’ ''b'') β ''x''}}. This is the analogue in monoid theory of a (left) [[Group action (mathematics)|group action]]. Right {{math|''M''}}-acts are defined in a similar way. A monoid with an act is also known as an ''[[operator monoid]]''. Important examples include [[transition system]]s of [[semiautomata]]. A [[transformation semigroup]] can be made into an operator monoid by adjoining the identity transformation.
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