Submonoid and abelian monoid Prove that - For any abelian monoid (M,*) ,the set of idempotent elements of M forms a submonoid ....
I can relate between idempotent elements and abelian monoid.. but how do we prove it as a submonoid?
 A: Let $M$ be a monoid with. For simplicity, I will write the operation by contatenation (as we do for multiplication), and the unit of $M$ as $1$. A submonoid of $M$ is a subset $A\subseteq M$ satisfying the following two properties:


*

*$1\in A$ (that is, $A$ contains the unit of $M$);

*If $x,y\in A$, then $xy\in A$ as well.


So the question is asking you to prove that the set of idempotents of $M$ is a submonoid of $M$. There are again some definitions to unravel here: An element $e$ of $M$ is idempotent if $ee=e$.
Good, now we have all definitions spelled out just in terms of the monoid structure.
Let $E=\left\{e\in M:ee=e\right\}$, be the set of idempotents of $M$. To check that $E$ is a submonoid of $M$, we need to verify that $E$ satisfies properties $1$ and $2$ above.


*

*To check that $1\in E$, we need to verify that satisfies the property defining $E$, that is, $11=1$. Do you see why this is true?

*Take arbitrary elements $x,y\in E$. Then, by hypothesis, $xx=x$ and $yy=y$, and $xy=yx$, because $M$ is commutative. We need to verify that $xy\in E$ as well, that is, that $(xy)(xy)=(xy)$. Let us try to write down a sequence of identities starting with the first term and ending with the second one. I'll start for you: By associativity of the operation:
$$(xy)(xy)=x(yx)y$$
can you use the hypotheses above to rewrite $x(yx)y$ as $xy$?
