Definition:Pointwise Inverse

Definition
Let $\struct {G, \oplus}$ be a group whose identity is $e_G$.

Let $S$ be a set.

Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\oplus$.

Let $f \in G^S$.

The pointwise inverse of $f$ in $G^S$ is denoted $f^*$ and defined as:


 * $\forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$

Also denoted as
If the operation on $G$ is related to or derived from addition, then $\map {f^*} x$ is often written as $\map {\paren {-f} } x$.

If the operation on $G$ is related to or derived from multiplication, then $\map {f^*} x$ is often written as $\map {\paren {f^{-1} } } x$ or $\map {\dfrac 1 f} x$.

Beware not to confuse $\map {\paren {f^{-1} } } x$ with the inverse mapping $\map {f^{-1} } x$.

Also see

 * Pointwise Inverse in Induced Structure