Answer to Question #162308 in Abstract Algebra for K

Question

Suppose * is an associative and commutative binary operation of a set S. Show that H = { a is an element of S | a*a=a} is closed under the operation *. 

Answer 1

Let $*$ be an associative commutative binary operation on a set $S$. Let us show that $H=\{a\in S\ |\ a*a=a\}$ is closed under operation $*.$ Let $a,b\in H$. Then $a*a=a$ and $b*b=b$ . It follows from commutativity and associativity of $*$ that$(a*b)*(a*b)= a*(b*(a*b))= a*((b*a)*b)= a*((a*b)*b)= a*(a*(b*b))= a*(a*b)= (a*a)*b=a*b.$

Therefore, $a*b\in H$, and $H$ is closed under operation $*.$