Definition:Zero (Number)/Naturally Ordered Semigroup

Definition
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

Then from NO 1, $\left({S, \circ, \preceq}\right)$ has a smallest element.

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

That is:
 * $\forall n \in S: 0 \preceq n$

Also see
It is proved in Zero is Identity in Naturally Ordered Semigroup that this element $0$ is the identity for $\circ$.

That is:
 * $\forall n \in S: n \circ 0 = n = 0 \circ n$