Invertible Elements under Natural Number Multiplication

Theorem
Let $\left({S, \circ, *, \preceq}\right)$ be a naturally ordered semigroup with product.

Let $m, n, p \in \left({S, \circ, *, \preceq}\right)$.

Then the only invertible element of $\left({S, *, \preceq}\right)$ is $1$.

Proof
Suppose $m \in S$ is invertible for $*$.

Let $n \in S: m * n = 1$.

Then from Multiplication in Naturally Ordered Semigroup has No Proper Zero Divisors:
 * $m \ne 0$ and $n \ne 0$

Thus, $1 \preceq m$ and $1 \preceq n$.

If $1 \prec m$ then from Ordering on Naturally Ordered Semigroup Product:
 * $1 \preceq n \prec m * n$

This contradicts $m * n = 1$.

The result follows.