Let $G$ be a semigroup, e.g., a set with an associative binary operation. Suppose further that $G$ has the divisibility property, e.g., for all $x,y\in G$ there exist $\ell,r\in G$ such that $\ell x=y$ and $xr=y$.
How can we prove that $\ell$ and $r$ are unique?
Ignoring associativity, this is almost the definition of a quasigroup; it only lacks the assumption that the $\ell$ and $r$ above are unique. On the nLab page for quasigroups, the following claim is made:
Note that we must specify, in the definition, that $\ell$ and $r$ are unique; without associativity, we cannot prove this.
So the implication is that with associativity, one can prove the uniqueness of $\ell$ and $r$. But I don't see how this follows.
To prove that $\ell$ is unique, it suffices to prove that the semigroup is right cancellative ($a x = b x$ implies $a = b$), since then $\ell_1 x = y$ and $\ell_2 x = y$ implies that $\ell_ 1 x = \ell_2 x$ and $\ell_1 = \ell_2$.
We first show that the semigroup has a right unit. Consider any element $a$. There is some $e$ such that $a e = a$. I claim that $b e = b$ for any $b$. This is because there is some $x$ such that $x a = b$, so $b e = x a e = x a = b$.
Now if $a x = b x$, there is some $y$ such that $x y = e$, so $a = a e = a x y = b x y = b e = b$.