Let $S$ be a semigroup. If for any $x,y\in S$, $x^2y=y=yx^2$, then prove that $S$ is an abelian group

Question

Let $S$ be a semigroup. If for any $x,y\in S$, $x^2y=y=yx^2$, then prove that $S$ is an abelian group.

My solution goes like this:

If for any $x,y\in S$, we have $x^2y=y=yx^2$. Then this implies $x^2=e$. So, $x=x^{-1}$ and this holds $\forall x\in S$. Hence, if $a,b\in S$ then $(a\cdot b)\in S$ and $$(a\cdot b)=(a\cdot b)^{-1}=b^{-1}a^{-1}=ba.$$ Thus, $S$ is an abelian group.

Is the above solution correct? Is it valid? If not, then where does it go wrong?

Answer 1

Your supposed proof assumes that there is an identity in the semigroup, which isn't always a given; you have to show it. See here for a real proof.