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Homesemigroup of transformations

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# 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 calculate 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

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

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