Ordering Compatible with Group Operation is Strongly Compatible/Corollary
Corollary to Ordering Compatible with Group Operation is Strongly Compatible
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group with identity $e$.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
The following hold:
\(\ds \forall x, y \in G: \, \) | \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds e \preccurlyeq y \circ x^{-1}\) | |||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds e \preccurlyeq x^{-1} \circ y\) | ||||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds x \circ y^{-1} \preccurlyeq e\) | ||||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds y^{-1} \circ x \preccurlyeq e\) |
\(\ds \forall x, y \in G: \, \) | \(\ds x \prec y\) | \(\iff\) | \(\ds e \prec y \circ x^{-1}\) | |||||||||||
\(\ds x \prec y\) | \(\iff\) | \(\ds e \prec x^{-1} \circ y\) | ||||||||||||
\(\ds x \prec y\) | \(\iff\) | \(\ds x \circ y^{-1} \prec e\) | ||||||||||||
\(\ds x \prec y\) | \(\iff\) | \(\ds y^{-1} \circ x \prec e\) |
Proof 1
By the definition of an ordered group, $\preccurlyeq$ is a relation compatible with $\circ$.
Thus by Relation Compatible with Group Operation is Strongly Compatible: Corollary, we obtain the first four results.
By Reflexive Reduction of Relation Compatible with Group Operation is Compatible, $\prec$ is compatible with $\circ$.
Again by Relation Compatible with Group Operation is Strongly Compatible: Corollary, we obtain the remaining results.
$\blacksquare$
Proof 2
Each result follows from Ordering Compatible with Group Operation is Strongly Compatible.
For example, by Ordering Compatible with Group Operation is Strongly Compatible:
- $x \preccurlyeq y \iff x \circ x^{-1} \preccurlyeq y \circ x^{-1}$
Since $x \circ x^{-1} = e$:
- $x \preccurlyeq y \iff e \preccurlyeq y \circ x^{-1}$
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups: Theorem $15.3$