## Definition

Let $S$ be a non-empty set, and let $\left({G, \circ}\right)$ 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:

$\forall f,g \in G^S: \forall s \in S: \left({f \circ g}\right) \left({s}\right) := f \left({s}\right) \circ g \left({s}\right)$

The double use of $\circ$ is justified as $\left({G^S, \circ}\right)$ inherits all abstract-algebraic properties $\left({G, \circ}\right)$ might have.

This is rigorously formulated and proved on Mappings to Algebraic Structure form Similar Algebraic Structure.

### Pointwise Multiplication

Let $\circ$ be used with multiplicative notation.

Then the operation defined above is called pointwise multiplication instead.