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.
$\blacksquare$