Definition:Subtraction/Naturally Ordered Semigroup
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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
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers