Axiom:Product Inverse Operation Axioms

Definition

Let $\struct {G, \oplus}$ be a closed algebraic structure on which the following hold:

$(\text {PI} 1)$	$:$	Self-Inverse Property	$\ds \forall x \in G:$	$\ds x \oplus x = e $
$(\text {PI} 2)$	$:$	Right Identity	$\ds \exists e \in G: \forall x \in G:$	$\ds x \oplus e = x $
$(\text {PI} 3)$	$:$	Product Inverse with Right Identity	$\ds \forall x, y \in G:$	$\ds e \oplus \paren {x \oplus y} = y \oplus x $
$(\text {PI} 4)$	$:$	Cancellation Property	$\ds \forall x, y, z \in G:$	$\ds \paren {x \oplus z} \oplus \paren {y \oplus z} = x \oplus y $

These four stipulations are known as the product inverse operation axioms.

The term product inverse operation was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.7 \ \text {(b)}$

\((\text {PI} 1)\)	$:$	Self-Inverse Property	\(\ds \forall x \in G:\)	\(\ds x \oplus x = e \)
\((\text {PI} 2)\)	$:$	Right Identity	\(\ds \exists e \in G: \forall x \in G:\)	\(\ds x \oplus e = x \)
\((\text {PI} 3)\)	$:$	Product Inverse with Right Identity	\(\ds \forall x, y \in G:\)	\(\ds e \oplus \paren {x \oplus y} = y \oplus x \)
\((\text {PI} 4)\)	$:$	Cancellation Property	\(\ds \forall x, y, z \in G:\)	\(\ds \paren {x \oplus z} \oplus \paren {y \oplus z} = x \oplus y \)