Ordering of Inverses in Ordered Monoid

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

 * First, to show that $x \prec y \implies y^{-1} \prec x^{-1}$:


 * Next, to show that $y^{-1} \prec x^{-1} \implies x \prec y$: