Difference in Naturally Ordered Semigroup is Unique

Theorem

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Let $n, m \in S$ such that $m \preceq n$.

Then there exists a unique difference $n - m$ of $m$ and $n$.

Proof

Since $m \preceq n$, by Naturally Ordered Semigroup Axiom $\text {NO} 3$: Existence of Product:

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

Now suppose that $p, q \in S$ are such that:

$m \circ p = m \circ q = n$

Then it follows from Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability that:

$p = q$

Hence the result.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers