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:

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

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