Ordering Compatible with Group Operation is Strongly Compatible/Corollary

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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

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