# 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:

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