Finding unity and units of a binary operation

Question

Let $\ast$ be the binary operation defined on $\mathbb{N}$ by $m \ast n = \max (m, n)$; the largest of $m$ and $n$. Decide whether unity exists and if so, find the units.

I know that unity is defined as $m * e=e * m=m$ so then $\max(m,e)=m$ which would be possible for all $m>e$ right? Does that mean that is the unity? Or is that the units? Thank you!

Answer 1

Hint: suppose $e$ exists; then $0*e=0$, so you have not many choices for $e$, do you? Next show that the candidate is indeed the neutral element.

Answer 2

Hint: Let $(X,\leq)$ be any linearly ordered set with minimum element $0$. Defining $x\ast y = max\{x,y\}$ will always yield a unique unit. Which one?