Prove the following properties of given semigroup.

Question

Let ({a,b},*) be a semigroup where $ a * a = b$. Show that:
(a) $ a*b=b*a$
(b) $b*b=b$

I've managed to prove the (a) part as: $$ a * b = a * ( a * a ) = (a * a) * a = b * a $$ But for the (b) part all I could work out was that if: $$ a * b = a = b * a $$ then $$\begin{align*} b*b&=(a*a)*(a*a)\\ &=(a*(a*b))*(a*a)\\ &=(b*b)*(a*a)\\ &=(b*a)*(b*a)\\ &=a*a=b \end{align*}$$ But I couldn't find a way to constructively find why $b*b = b$. Any help would be greatly appreciated.

Answer 1

If $a*b \ne a$, then $a*b=b$ and $b*b=a*a*b=a*b=b$.

Answer 2

The hint:

If $ab=b$ then $a^2=b^2=ab=ba=b$.

If $ab=a$ we get $b^2=b$ again.