Inverse not always Unique for Non-Associative Operation

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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 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:

\(\ds \paren {a \circ a} \circ b\) \(=\) \(\ds e \circ b\)
\(\ds \) \(=\) \(\ds b\)

\(\ds a \circ \paren {a \circ b}\) \(=\) \(\ds a \circ e\)
\(\ds \) \(=\) \(\ds a\)

and so $\circ$ is not associative.

Note further that:

\(\ds a \circ b\) \(=\) \(\ds e\)
\(\ds \) \(=\) \(\ds b \circ a\)

and also:

\(\ds a \circ a\) \(=\) \(\ds e\)

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

Hence the result.


Also see