Transformation Semigroup

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In algebra, a transformation semigroup (or composition semigroup) is a collection of functions from a set to itself which is closed under function composition. If it includes the identity function, it is a transformation (or composition) monoid. This is the semigroup anologue of a permutation group. A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being effective, i.e., if two elements of the semigroup have the same action, then they are equal. An analogue of Cayley''s theorem shows that any semigroup can be realized as a transformation semigroup of some set.