Strict Ordering of Naturally Ordered Semigroup is Strongly Compatible
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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
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.3$