Ordering of Naturally Ordered Semigroup is Strongly Compatible

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Theorem

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


Then $\preceq$ is strongly compatible with $\circ$:

$\forall m, n, p \in S: m \preceq n \iff m \circ p \preceq n \circ p$


Proof

The forward implication is immediate from $\preceq$ being compatible with $\circ$:

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


Conversely, suppose that $m \circ p \preceq n \circ p$.

Suppose that $n \prec m$.

Then as $\preceq$ is compatible with $\circ$:

$n \circ p \preceq m \circ p$

Since $\preceq$ is an ordering, this implies:

$n \circ p = m \circ p$

By Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability, it follows that:

$n = m$

contradicting our assumption that $n \prec m$.


Hence, since $\preceq$ is a total ordering:

$m \preceq n$

as desired.

$\blacksquare$


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