Definition:Zero (Number)/Naturally Ordered Semigroup

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

Then from axiom $(\text {NO} 1)$, $\struct {S, \circ, \preceq}$ 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