Unique Minus

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Theorem

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

Then:

$\forall m, n \in \left({S, \circ, \preceq}\right): m \preceq n \iff \exists! p \in S: m \circ p = n$

That is, for any two elements in $S$, there is a unique element which forms the sum.

So we define the operation $\ominus$, as follows:

$\forall m, n \in \left({S, \circ, \preceq}\right), m \preceq n: n \ominus m = p := m \circ p = n$

That is, $n \ominus m = p$ is the unique element $p$ such that $m \circ p = n$.

The operation $\ominus$ is called (in this context) minus.

$\forall m, n \in S: m \preceq n \implies \exists p \in S: m \circ p = n$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle m \circ p$	$=$	$\displaystyle n$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle m = m \circ 0$	$\preceq$	$\displaystyle m \circ p = n$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Cancellability in Naturally Ordered Semigroup
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle m$	$\preceq$	$\displaystyle n$	$\displaystyle $	$\displaystyle $	$\displaystyle $

So $\forall m, n \in S: m \preceq n \iff \exists p \in S: m \circ p = n$.

From naturally ordered semigroup: NO 2, we have $m \circ q = n = m \circ p \implies q = p$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle m \circ p\)	\(=\)	\(\displaystyle n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle m = m \circ 0\)	\(\preceq\)	\(\displaystyle m \circ p = n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Cancellability in Naturally Ordered Semigroup
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle m\)	\(\preceq\)	\(\displaystyle n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)