Definition:Unit (One)/Naturally Ordered Semigroup

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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 Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements, $\struct {S^*, \circ, \preceq}$ has a smallest element for $\preceq$.

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