Editing Semigroup action (section)

{{Short description|An action or act of a semigroup on a set}}
{{Redirect|S-set|the suburban train fleet|Sydney Trains S set}}

In [[algebra]] and [[theoretical computer science]], an '''action''' or '''act''' of a '''[[semigroup]]''' on a [[Set (mathematics)|set]] is a rule which associates to each element of the semigroup a [[transformation (geometry)|transformation]] of the set in such a way that the product of two elements of the semigroup (using the semigroup [[binary operation|operation]]) is associated with the [[function composition|composite]] of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are ''acting'' as transformations of the set. From an [[algebraic structure|algebraic]] perspective, a semigroup action is a generalization of the notion of a [[Group action (mathematics)|group action]] in [[group (mathematics)|group theory]]. From the computer science point of view, semigroup actions are closely related to [[finite-state machine|automata]]: the set models the state of the automaton and the action models transformations of that state in response to inputs.

An important special case is a '''monoid action''' or '''act''', in which the semigroup is a [[monoid]] and the [[identity element]] of the monoid acts as the [[identity transformation]] of a set. From a [[category theoretic]] point of view, a monoid is a [[category (mathematics)|category]] with one object, and an act is a functor from that category to the [[category of sets]]. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets.

Another important special case is a '''[[transformation semigroup]]'''. This is a semigroup of transformations of a set, and hence it has a tautological action on that set. This concept is linked to the more general notion of a semigroup by an analogue of [[Cayley's theorem]].

''(A note on terminology: the terminology used in this area varies, sometimes significantly, from one author to another. See the article for details.)''