Inverse in Monoid is Unique

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Theorem

Let $\struct {S, \circ}$ be a monoid.


Then an element $x \in S$ can have at most one inverse for $\circ$.


Proof

Let $e$ be the identity element of $\struct {S, \circ}$.

Suppose $x \in S$ has two inverses: $y$ and $z$.


Then:

\(\ds y\) \(=\) \(\ds y \circ e\) Definition of Identity Element
\(\ds \) \(=\) \(\ds y \circ \paren {x \circ z}\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds \paren {y \circ x} \circ z\) Monoid Axiom $\text S 1$: Associativity
\(\ds \) \(=\) \(\ds e \circ z\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds z\) Definition of Identity Element


Similarly:

\(\ds y\) \(=\) \(\ds e \circ y\) Definition of Identity Element
\(\ds \) \(=\) \(\ds \paren {z \circ x} \circ y\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds z \circ \paren {x \circ y}\) Monoid Axiom $\text S 1$: Associativity
\(\ds \) \(=\) \(\ds z \circ e\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds z\) Definition of Identity Element

So whichever way round you do it, $y = z$ and the inverse of $x$ is unique.

$\blacksquare$


Also see


Sources

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