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 ${𝒯}_X$.

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

When $X$ is finite, say $X=\{x_1,x_2,\mathrm{\dots },x_n\}$, 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 products. For example, if $X=\{1,2,3,4\}$, then

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