Definition:Operation Induced by Injection

From ProofWiki
Jump to: navigation, search

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 $f^* \circ$.