Elements and operation of a monoid

Question

Let $(X, \star)$ be a monoid, $e\in X$ the identity element and let $x\star x=e$ for all $x\in X$.

Show the following:

1. for all $x\in X$ : $x$ is invertible
2. $\star$ is commutative

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Could you give me a hint for 1 ? We have that $x\star x=e$ Doesn't this means that $x$ has itself as the inverse element? $$$$

As for 2. I have done the following:

Let $x,y\in X$. Thn we get $$x\star x=e \Rightarrow y\star x\star x=y\star e \Rightarrow y\star x\star x=y \Rightarrow y\star x\star x\star x=y\star x \Rightarrow y\star x\star e=y\star x \Rightarrow y\star x=y\star x$$which means that $\star$ is commutative.

Is this proof correct?

Answer 1

You're right about 1. For 2, I'll hide the stars. Since $xyyx=xx=e=xyxy$, the result follows from $xy$ being invertible.