Definition:Pointwise Operation on Integer-Valued Functions

Definition
Let $S$ be a set.

Let $\Z^S$ be the set of all mappings $f: S \to \Z$, where $\Z$ is the set of integers.

Let $\oplus$ be a binary operation on $\Z$.

Define $\oplus: \Z^S \times \Z^S \to \Z^S$, called pointwise $\oplus$, by:


 * $\forall f, g \in \Z^S: \forall s \in S: \left({f \oplus g}\right) \left({s}\right) := f \left({s}\right) \oplus g \left({s}\right)$

In the above expression, the operator on the RHS is the given $\oplus$ on the integers.

Specific Instantiations
When $\oplus$ has a specific name, for example "addition" or "multiplication", it is usual to name the corresponding pointwise operation by prepending pointwise to that name.

Also see
It can be seen that these definitions instantiate the more general Pointwise Operation on Number-Valued Functions.