# 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

### 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 \ \$ $\displaystyle x$ $\prec$ $\displaystyle y$

$\blacksquare$