# Definition:Operation Induced on Set of Mappings

## Contents

## 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 $\C$.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 13$