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

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