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 $$y^{-1} \prec x^{-1} \implies x \prec y$$:

$$ $$ $$