Identity of Inverse Completion of Commutative Monoid

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Theorem

Let $\struct {S, \circ}$ be a commutative monoid whose identity is $e$.

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 $e \in T$ is the identity for $\circ'$.


Proof

Let $e$ be the identity for $\circ$.

Let $e = x \circ' y^{-1}$, where $x \in S, y \in C$.


Then:

\(\ds y\) \(=\) \(\ds e \circ y\) Definition of Identity Element
\(\ds \) \(=\) \(\ds \paren {x \circ' y^{-1} } \circ' y\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds x \circ' \paren {y^{-1} \circ' y}\) $\circ'$ is associative
\(\ds \) \(=\) \(\ds x \circ' e\) Definition of Inverse Element


Thus $e = y^{-1} \circ' y$, and $y^{-1} \circ' y$ is the identity for $\circ'$.

$\blacksquare$


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