Hw2 Problem 13

### Problem 13

(*) Suppose that $*$ is an associative and commutative binary operation on a set $S$. Show that $H = \{a \in S\ |\ a * a = a\}$ is closed under $*$. (The elements of H are idempotents of the binary operation $*$.)

Solution

Let $a, b \in H$, then $a * a = a$ and $b * b = b$. Show $(a * b) * (a * b) = (a * b)$, then $(a * b) \in H$. Using associativity and commutativity:

(1)
\begin{align} (a * b) * (a * b)= [(a * b) * a] * b = [a * (b * a)] * b\\= [a * (a * b)] * b = [(a * a) * b] * b\\= (a * b) * b = a * (b * b)\\= (a * b) \end{align}

Therefore $H$ is closed under $*$.