Definition:Pointwise Operation

Definition
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 $\circ$.

Also known as
A pointwise operation is also referred to as an induced operation.

It is usual to use the same symbol for the pointwise operation as for the operation that induces it.

Thus one would refer to the structure on $T^S$ induced by $\circ$ as $\struct {T^S, \circ}$.

In most reference works, the precise properties of a pointwise operation are taken to be implicitly inherited from its base operation.

Also see

 * Definition:Pointwise Operation on Number-Valued Functions: this definition crystallises when $T$ is taken to be one of the standard number sets $\N, \Z, \Q, \R$ and $\C$.