Ordering in terms of Addition

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Theorem

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

Then $\forall m, n \in S$:

$m \preceq n \iff \exists p \in S: m \circ p = n$


Proof

Necessary Condition

From axiom $(NO3)$, we have:

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

$\Box$


Sufficient Condition

Suppose that $m \circ p = n$.

\(\displaystyle 0\) \(\preceq\) \(\displaystyle p\) Definition of Zero
\(\displaystyle \implies \ \ \) \(\displaystyle m \circ 0\) \(\preceq\) \(\displaystyle m \circ p\) $\preceq$ is compatible with $\circ$
\(\displaystyle \implies \ \ \) \(\displaystyle m\) \(\preceq\) \(\displaystyle n\) Zero is Identity in Naturally Ordered Semigroup

$\Box$


So $\forall m, n \in S$:

$m \preceq n \iff \exists p \in S: m \circ p = n$

$\blacksquare$


Sources