Definition:Zero (Number)

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