Ordering Compatible with Group Operation is Strongly Compatible/Corollary/Proof 2
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Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group with identity $e$.
Let $\prec$ be the reflexive reduction of $\preccurlyeq$.
Let $x, y \in G$.
Then the following equivalences 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
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 $x \circ x^{-1} = e$:
- $x \preccurlyeq y \iff e \preccurlyeq y \circ x^{-1}$
$\blacksquare$