Ordering Compatible with Group Operation is Strongly Compatible
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Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group whose identity element is $e$.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
The following hold:
\(\ds \forall x, y, z \in G: \, \) | \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds x \circ z \preccurlyeq y \circ z\) | |||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds z \circ x \preccurlyeq z \circ y\) | ||||||||||||
\(\ds x \prec y\) | \(\iff\) | \(\ds x \circ z \prec y \circ z\) | ||||||||||||
\(\ds x \preceq y\) | \(\iff\) | \(\ds z \circ x \prec z \circ y\) |
Corollary
\(\ds \forall x, y \in G: \, \) | \(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds e \preccurlyeq y \circ x^{-1}\) | |||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds e \preccurlyeq x^{-1} \circ y\) | ||||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds x \circ y^{-1} \preccurlyeq e\) | ||||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds y^{-1} \circ x \preccurlyeq e\) |
\(\ds \forall x, y \in G: \, \) | \(\ds x \prec y\) | \(\iff\) | \(\ds e \prec y \circ x^{-1}\) | |||||||||||
\(\ds x \prec y\) | \(\iff\) | \(\ds e \prec x^{-1} \circ y\) | ||||||||||||
\(\ds x \prec y\) | \(\iff\) | \(\ds x \circ y^{-1} \prec e\) | ||||||||||||
\(\ds x \prec y\) | \(\iff\) | \(\ds y^{-1} \circ x \prec e\) |
Proof
By definition of ordered group, $\preccurlyeq$ is a relation compatible with $\circ$.
Thus by Relation Compatible with Group Operation is Strongly Compatible:
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds x \circ z \preccurlyeq y \circ z\) | ||||||||||||
\(\ds x \preccurlyeq y\) | \(\iff\) | \(\ds z \circ x \preccurlyeq z \circ y\) |
By Reflexive Reduction of Relation Compatible with Group Operation is Compatible, $\prec$ is compatible with $\circ$.
Thus again by Relation Compatible with Group Operation is Strongly Compatible:
\(\ds x \prec y\) | \(\iff\) | \(\ds x \circ z \prec y \circ z\) | ||||||||||||
\(\ds x \preceq y\) | \(\iff\) | \(\ds z \circ x \prec z \circ y\) |
Hence the result.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups: Theorem $15.3$