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Is it true that inverse of an element $a\in S$, a semigroup is unique? It seems like this should be logical but I just can't get it.

Any help is greatly appreciated.

[For a an elements $a,b$ in semigroup $S$, we say $b$ is inverse of $a$ if we have $bab=b$ and $aba=a$.]

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    $\begingroup$ Note: While inverses of a semigroup are generally not unique, the common definition of an inverse semigroup demands that inverse of every element exist and is unique. $\endgroup$
    – Jakobian
    Jun 13, 2023 at 12:16

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What about $\mathbb{Z}^2$ with the operation $(a,b)\cdot (c,d) = (a+c,b)$?

Then $(1,1)$ has the inverse $(-1,n)$ for all $n$.

And in general, an inverse of $(i,j)$ is given by $(-i,n)$ for any $n$.

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