Definition:Induced Structure

Definition
Let $\left({T, \oplus}\right)$ be an algebraic structure, and let $S$ be a set.

Let $T^S$ be the set of all mappings from $S$ to $T$.

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

Let $f, g \in T^S$, that is, let $f: S \to T$ and $g: S \to T$ be mappings.

Then $f \oplus g$ is defined as:


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

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