# Definition:Pointwise Operation/Induced Structure

< Definition:Pointwise Operation(Redirected from Definition:Induced Structure)

Jump to navigation
Jump to search
## 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.

Let $f \oplus g$ be the **pointwise 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

The **algebraic structure on $T^S$ induced by $\circ$** is also referred to just as the **induced structure**.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces