Definition:Pointwise Operation/Induced Structure
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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