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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Theorem $13.6: \ 3^\circ$