\begin{definition}[Definition:Pointwise Addition of Mappings]
Let $S$ be a non-empty set.
Let $\struct {G, \circ}$ be a commutative semigroup.
Let $G^S$ be the set of all mappings from $S$ to $G$.
Then '''pointwise addition''' on $G^S$ is the binary operation $\circ: G^S \times G^S \to G^S$ (the $\circ$ is the same as for $G$) defined by:
:$\forall f, g \in G^S: \forall s \in S: \map {\paren {f \circ g} } s := \map f s \circ \map g s$
The double use of $\circ$ is justified as $\struct {G^S, \circ}$ inherits all abstract-algebraic properties $\struct {G, \circ}$ might have.
This is rigorously formulated and proved on Mappings to Algebraic Structure form Similar Algebraic Structure.
\end{definition}