Definition:Unit (One)/Naturally Ordered Semigroup
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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 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 $1$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers