Definition:Zero (Number)/Naturally Ordered Semigroup

Definition

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Then from Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered, $\struct {S, \circ, \preceq}$ has a smallest element.

This smallest element of $\struct {S, \circ, \preceq}$ is called zero and has the symbol $0$.

That is:

$\forall n \in S: 0 \preceq n$

Also see

• Zero is Identity in Naturally Ordered Semigroup

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers