Definition:Subtraction/Naturally Ordered Semigroup

Definition

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

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

By Naturally Ordered Semigroup Axiom $\text {NO} 3$: Existence of Product, there exists a $p \in S$ such that:

$m \circ p = n$

This $p$ is the difference between $m$ and $n$, and denoted $n - m$.

The operation $-$, assigning to $m, n \in S$ with $m \preceq n$ their difference $n - m$ is called subtraction.

Also see

• Difference in Naturally Ordered Semigroup is Unique

Sources

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