Inverse not always Unique for Non-Associative Operation

Theorem
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\circ$ be a non-associative operation.

Then for any $x \in S$, it is possible for $x$ to have more than one inverse element.

Proof
Proof by Counterexample:

Consider the algebraic structure $\struct {S, \circ}$ consisting of:
 * The set $S = \set {a, b, e}$
 * The binary operation $\circ$

whose Cayley table is given as follows:


 * $\begin {array} {c|cccc}

\circ & e & a & b \\ \hline e & e & a & b \\ a & a & e & e \\ b & b & e & e \\ \end {array}$

By inspection, we see that $e$ is the identity element of $\struct {S, \circ}$.

We also note that:

and so $\circ$ is not associative.

Note further that:

and also:

So both $a$ and $b$ are inverses of $a$.

Hence the result.

Also see

 * Inverse in Monoid is Unique