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:

\(\displaystyle y\) \(=\) \(\displaystyle y \circ e\) $\quad$ Definition of Identity Element $\quad$
\(\displaystyle \) \(=\) \(\displaystyle y \circ \paren {x \circ z}\) $\quad$ Definition of Inverse Element $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {y \circ x} \circ z\) $\quad$ Definition of Associative Operation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle e \circ z\) $\quad$ Definition of Inverse Element $\quad$
\(\displaystyle \) \(=\) \(\displaystyle z\) $\quad$ Definition of Identity Element $\quad$


Similarly:

\(\displaystyle y\) \(=\) \(\displaystyle e \circ y\) $\quad$ Definition of Identity Element $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {z \circ x} \circ y\) $\quad$ Definition of Inverse Element $\quad$
\(\displaystyle \) \(=\) \(\displaystyle z \circ \paren {x \circ y}\) $\quad$ Definition of Associative Operation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle z \circ e\) $\quad$ Definition of Inverse Element $\quad$
\(\displaystyle \) \(=\) \(\displaystyle z\) $\quad$ Definition of Identity Element $\quad$

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

$\blacksquare$


Also see


Sources