Proving a monoid is associative

Question

Let M be a commutative monoid and set $M^+=$ {$a \in M : a^k$ is idempotent for some $k\ge 1$}. Prove that $M^+$ is a monoid with the binary operation induced from M.

I have proven $M^+$ is closed and that it has an identity but I am now stuck on proving it is associative. Is it enough to show that $(a*b)*c=a*(b*c)$? Or do I have to show it using the $k$'s?

Answer 1

If you prove that $M^+$ has an identity and is closed under the binary operation, then you are done.

Answer 2

An associative binary operation on a set induces an associative binary operation on its subsets that are closed under the operation. In your case:

$$(a *_{M^+} b)*_{M^+}c = (a*_M b)*_M c = a*_M (b*_M c) = a *_{M^+} (b*_{M^+}c).$$