\begin{definition}[Definition:Pointwise Operation/Induced Structure]
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.
Let $f \oplus g$ be the '''pointwise operation on $T^S$ induced by $\circ$'''.
The algebraic structure $\struct {T^S, \oplus}$ is called the '''algebraic structure on $T^S$ induced by $\circ$'''.
\end{definition}