Question

Prove that the set of idempotent elements of a commutative monoid m, forms a submonoid.

Answer 1

Answer:

Let's call the set of idempotent elements A.

You just need to establish that the identity (let's call it e) is in A and that A is closed under multiplication.

The first bit is easy.  Since e is the identity, we have e*e = e, so it is idempotent, meaning that it is in A, the set of all idempotent elements.

Next, consider any two elements a, b in A.  To show that A is closed under multiplication means that we need to show that the product ab is in A, too.  This just means that we need to show that ab is idempotent, so we need to show that (ab)(ab) = ab.  But our monoid is commutative, so

(ab)(ab) = aabb

and since a and b are idempotent, we have aa = a and bb = b.  Thus

(ab)(ab) = ab,

as required.

Answer 2

Answer:

for any commutative monoid m