\begin{definition}[Definition:Operation Induced by Injection]
Let $\struct {T, \circ}$ be an algebraic structure.
Let $f: S \to T$ be an injection.
Then the '''operation induced on $S$ by $f$ and $\circ$''' is the binary operation $\circ_f$ on $S$ defined by:
:$\circ_f: S \times S \to S: x \circ_f y := \map {f^{-1} } {\map f x \circ \map f y}$
\end{definition}