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