Definition:Operation Induced by Injection

Definition
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:


 * $\circ_f: S \times S \to S: x \circ_f y := \map {f^{-1} } {\map f x \circ \map f y}$

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

Also see

 * Definition:Pullback of Quotient Group Isomorphism

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