\begin{definition}[Definition:Pointwise Operation]
Let $S$ be a set.
Let $\struct {T, \circ}$ be an algebraic structure.
Let $T^S$ be the set of all mappings from $S$ to $T$.
Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.
Then the operation $f \oplus g$ is defined on $T^S$ as follows:
:$f \oplus g: S \to T: \forall x \in S: \map {\paren {f \oplus g} } x = \map f x \circ \map g x$
The operation $\oplus$ is called the '''pointwise operation on $T^S$ induced by $\circ$'''.
\end{definition}