Definition:Pointwise Operation/Induced Structure

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