Ordering in terms of Addition

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

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


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

Necessary Condition
From axiom $(NO3)$, we have:


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

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


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