Definition:Subtraction/Naturally Ordered Semigroup

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

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

By axiom $(NO3)$, there exists a $p \in S$ such that:


 * $m + p = n$

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