Ordering of Inverses in Ordered Monoid

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Theorem

Let $\left({S, \circ, \preceq}\right)$ be an ordered monoid whose identity is $e$.

Let $x, y \in S$ be invertible.


Then:

$x \prec y \iff y^{-1} \prec x^{-1}$


Proof

Necessary Condition

\(\displaystyle x\) \(\prec\) \(\displaystyle y\)
\(\displaystyle \implies \ \ \) \(\displaystyle e\) \(=\) \(\displaystyle x^{-1} \circ x \prec x^{-1} \circ y\) Strict Ordering Preserved under Product with Cancellable Element
\(\displaystyle \implies \ \ \) \(\displaystyle y^{-1}\) \(=\) \(\displaystyle e \circ y^{-1} \prec x^{-1} \circ y \circ y^{-1} = x^{-1}\)
\(\displaystyle \implies \ \ \) \(\displaystyle y^{-1}\) \(\prec\) \(\displaystyle x^{-1}\)

$\Box$


Sufficient Condition

\(\displaystyle y^{-1}\) \(\prec\) \(\displaystyle x^{-1}\)
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle \left({x^{-1} }\right)^{-1} \prec \left({y^{-1} }\right)^{-1} = y\)
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(\prec\) \(\displaystyle y\)

$\blacksquare$


Sources