A monoid is a set with an operation that is associative and has an identity element.

Any finite monoid can be represented as a set of functions from a finite set to itself.

For any natural k, T_k is the monoid of *all *functions from the set {1,...,k} to itself, where the operation is composition of functions. There are k^k elements in this monoid, and the identity element is the identity function.

A monoid homomorphism is a function f from one monoid M to another monoid N, with the property that f(xy) = f(x)f(y) for any two elements x and y of M.

-- BarrinG - 2012-04-30

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