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:

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

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