Inverse Element in Inverse Completion of Commutative Monoid
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Theorem
Let $\struct {S, \circ}$ be a commutative monoid.
Let $\struct {C, \circ} \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$.
Let $\struct {T, \circ'}$ be an inverse completion of $\struct {S, \circ}$.
Then the inverse of an element of $S$ which is invertible for $\circ$ is also its inverse for $\circ'$.
Proof
Let the identity of $\struct {S, \circ}$ be $e$.
Let $z$ be the inverse of $y$ for $\circ$:
- $z \circ y = e$
- $y \circ z = e$
From Identity of Inverse Completion of Commutative Monoid:
- $z \circ' y = e$
- $y \circ' z = e$
Hence $z$ is the inverse of $y$ for $\circ'$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers: Theorem $20.1: \ 5^\circ$