Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible

Theorem

Let $\struct {S, \circ, \preceq}$ 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 Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability, 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 Naturally Ordered Semigroup Axiom $\text {NO} 1$: Well-Ordered, $\preceq$ 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 Naturally Ordered Semigroup Axiom $\text {NO} 2$: Cancellability.

The result follows.

$\blacksquare$

Also see

• Ordering of Naturally Ordered Semigroup is Strongly Compatible

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.3$