Definition:Zero (Number)

Definition
The number zero is defined as being the cardinal of the empty set.

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

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

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

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

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

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