semigroup of transformations

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

If $\alpha$ and $\beta$ are transformations on $X$ , then their product $\alpha\beta$ is defined (writing functions on the right) by $(x)(\alpha\beta)=((x)\alpha)\beta$ .

With this definition, the set of all transformations on $X$ becomes a semigroup, the full semigroupf of transformations on $X$ , denoted $\mathcal{T}_{X}$ .

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

When $X$ is finite, say $X=\{x_{1},x_{2},\dots,x_{n}\}$ , then the transformation $\alpha$ which maps $x_{i}$ to $y_{i}$ (with $y_{i}\in X$ , of course) is often written:

$\alpha=\begin{pmatrix}x_{1}&x_{2}&\dots&x_{n}\\ y_{1}&y_{2}&\dots&y_{n}\end{pmatrix}$

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

$\begin{pmatrix}1&2&3&4\\ 3&2&1&2\end{pmatrix}\begin{pmatrix}1&2&3&4\\ 2&3&3&4\end{pmatrix}=\begin{pmatrix}1&2&3&4\\ 3&3&2&3\end{pmatrix}$

When $X$ is infinite, say $X=\{1,2,3,\ldots\}$ , then this notation is still useful for illustration in cases where the transformation pattern is apparent. For example, if $\alpha\in\mathcal{T}_{X}$ is given by $\alpha\colon n\mapsto n+1$ , we can write

$\alpha=\begin{pmatrix}1&2&3&4&\dots\\ 2&3&4&5&\dots\end{pmatrix}$

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