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 operation on $T^S$ induced by $\circ$.

The algebraic structure $\struct {T^S, \oplus}$ is called the algebraic structure on $T^S$ induced by $\circ$.

Also known as
It is usual to use the same symbol for the induced operation as for the operation that induces it.

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

Operations of this type are often referred to as pointwise operations.

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

Also see

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