Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible

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Theorem

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


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

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


Proof

By axiom $(NO2)$, all $n \in S$ are cancellable.

Hence from Strict Ordering Preserved under Product with Cancellable Element:

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


By axiom $(NO1)$, $\preceq$ is a total ordering.

Therefore, the contrapositive of:

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

is:

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

which we know to be true by virtue of axiom $(NO2)$.


The result follows.

$\blacksquare$


Also see


Sources