Product Inverse Operation Properties
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Theorem
Let $\struct {G, \oplus}$ be a closed algebraic structure on which the following properties 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.
Let $\circ$ be the operation on $G$ defined as:
- $\forall x, y \in G: x \circ y = x \oplus \paren {e \oplus y}$
The following lemmata hold:
Lemma 1
- $\forall x, y, z \in G: \paren {x \circ z} \oplus \paren {y \circ z} = x \oplus y$
Lemma 2
- $\forall x, y, z \in G: \paren {x \oplus z} \circ \paren {z \oplus y} = x \oplus y$
Lemma 3
- $\forall x, y \in G: \quad x \oplus y = e \implies x = y$
Lemma 4
- $\forall x, y, z \in G: x \oplus z = y \oplus z \implies x = y$
Lemma 5
- $\forall x, y \in G: \paren {x \circ y} \oplus y = x$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.7 \ \text {(b)}$