Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup
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Theorem
Let $\struct {S, \circ, \preceq}$ be a totally ordered semigroup.
If either:
- $x \circ z \prec y \circ z$
or
- $z \circ x \prec z \circ y$
then $x \prec y$.
Proof
Let $x \circ z \prec y \circ z$.
Aiming for a contradiction, suppose $x \succeq y$.
As $\struct {S, \circ, \preceq}$ is an ordered semigroup, $\preceq$ is compatible with $\circ$.
Hence we have:
- $x \succeq y \implies x \circ z \succeq y \circ z$
which contradicts $x \circ z \prec y \circ z$.
We have that $\preceq$ is a total ordering, and that it is not the case that $x \succeq y$.
Hence by the Trichotomy Law:
- $x \prec y$
Similarly for $z \circ x \prec z \circ y$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups: Theorem $15.1: \ 2^\circ$