Definition:Unit (One)/Naturally Ordered Semigroup

Definition
Let $\struct {S, \circ, \preceq}$ 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 $(\text {NO} 4)$, it has a smallest element for $\preceq$.

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