Definition:Unit (One)/Naturally Ordered Semigroup

Definition
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

Let $S^*$ be the zero complement of $S$.

By Zero Complement is Not Empty, $S^*$ is not empty.

Therefore, by axiom $(NO4)$, it has a smallest element for $\preceq$.

This smallest element is called one and denoted $1$.