# Ordering in terms of Addition

## Theorem

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

Then $\forall m, n \in S$:

$m \preceq n \iff \exists p \in S: m \circ p = n$

## Proof

### Necessary Condition

From axiom $(NO3)$, we have:

$\forall m, n \in S: m \preceq n \implies \exists p \in S: m \circ p = n$

$\Box$

### Sufficient Condition

Suppose that $m \circ p = n$.

 $\displaystyle 0$ $\preceq$ $\displaystyle p$ Definition of Zero $\displaystyle \implies \ \$ $\displaystyle m \circ 0$ $\preceq$ $\displaystyle m \circ p$ $\preceq$ is compatible with $\circ$ $\displaystyle \implies \ \$ $\displaystyle m$ $\preceq$ $\displaystyle n$ Zero is Identity in Naturally Ordered Semigroup

$\Box$

So $\forall m, n \in S$:

$m \preceq n \iff \exists p \in S: m \circ p = n$

$\blacksquare$