# Difference in Naturally Ordered Semigroup is Unique

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## 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 axiom $(NO3)$:

- $\exists p \in S: m + p = n$

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

- $m + p = m + q = n$

Then it follows from axiom $(NO2)$ that:

- $p = q$

Hence the result.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 16$: Theorem $16.2$