Induced Structure Identity

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

Let $\left({T^S, \oplus}\right)$ be the structure on $T^S$ induced by $\circ$.

If $e$ is an identity for $\circ$, then the constant mapping $f_e: S \to T$ defined as:


 * $\forall x \in S: f_e \left({x}\right) = e$

is the identity for the operation induced on $T^S$ by $\circ$.

Proof
Let $\left({T, \circ}\right)$ be a structure with an identity $e$.

Let $f \in T^S$. Then: