Does the binary operation $m ⋆ n = m^n$ on $\mathbb N$ have a neutral element?

Question

Does the binary operation $\,m ⋆ n = m^n\,$ on $\,\mathbb N\,$ have a neutral element?

I said yes, and it is $\,e=1\,$ because $\,m ⋆ e = m^e = m^1 = m,\;$ but apparently that is wrong.

Answer 1

Recall that for a neutral element $e$, we must also have that $e\star m = m$.

Now, if $e = 1$ as you propose, then, unless $m = e = 1$

$$m\neq 1 \implies (e\star m = 1^m = 1\neq m)$$

Answer 2

Suppose $k$ is a neutral element.

Then $m\star k = m$ for all $m\in\mathbb{N}$.

But this means that $m^k = m$ for all $m\in\mathbb{N}$.

So $k=1$. However then $k\star m = m$ for all $m\in\mathbb{N}$ fails!