semigroup of transformations

Let X be a set. A transformationMathworldPlanetmath of X is a function from X to X.

If α and β are transformations on X, then their productPlanetmathPlanetmath αβ is defined (writing functions on the right) by (x)(αβ)=((x)α)β.

With this definition, the set of all transformations on X becomes a semigroup, the full semigroupf of transformations on X, denoted 𝒯X.

More generally, a semigroup of transformations is any subsemigroup of a full set of transformations.

When X is finite, say X={x1,x2,,xn}, then the transformation α which maps xi to yi (with yiX, of course) is often written:


With this notation it is quite easy to products. For example, if X={1,2,3,4}, then


When X is infiniteMathworldPlanetmath, say X={1,2,3,}, then this notation is still useful for illustration in cases where the transformation pattern is apparent. For example, if α𝒯X is given by α:nn+1, we can write

Title semigroup of transformations
Canonical name SemigroupOfTransformations
Date of creation 2013-03-22 13:07:36
Last modified on 2013-03-22 13:07:36
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 20M20
Synonym transformation semigroup
Defines full transformation semigroup