# Definition:Operation Induced by Injection

## Definition

Let $\left({T, \circ}\right)$ 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:

- $\circ_f: S \times S \to S: x \circ_f y := f^{-1} \left({f \left({x}\right) \circ f \left({y}\right)}\right)$

## Also known as

This is an instance of the more general phenomenon of a structure being pulled back along a function.

## Also see

Thus $\circ_f$ may be called the **pullback of $\circ$ along $f$**; it may be denoted by $f^* \circ$.