Given $a \ast a=a \forall a \in N$ on a set $N$ for an arbitrary binary operation $\ast$:
Let us define the binary operation as $a \ast b = a^2$. Clearly, $a^2 \neq a \forall a \in A$. Therefore the statement is false.
We can verify this easily. Let $a=b=2 \rightarrow a \ast b = 2^2 = 4$ which is $\neq a = 2$.