Definition:Zero (Number)/Naturally Ordered Semigroup

Definition

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

Then from axiom $(NO 1)$, $\left({S, \circ, \preceq}\right)$ 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

• Zero is Identity in Naturally Ordered Semigroup

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 16$