Definition:Operation Induced on Set of Mappings

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Definition

Let $\left({T, \circ}\right)$ be an algebraic structure.

Let $S$ be a set.

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:

$f \oplus g: S \to T: \forall x \in S: \left({f \oplus g}\right) \left({x}\right) = f \left({x}\right) \circ g \left({x}\right)$

The operation $\oplus$ is called the operation on $T^S$ induced by $\circ$.

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

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 $\left({T^S, \circ}\right)$.

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.

Pointwise Operation on Real-Valued Functions explains how this definition crystallises when $T$ is taken to be the set of real numbers $\R$