Inversion Mapping Reverses Ordering in Ordered Group

Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group.

Let $x, y \in G$.

Let $\prec$ be the reflexive reduction of $\preceq$.

Then the following equivalences hold:

Proof
By the definition of an ordered group, $\preceq$ is a relation compatible with $\circ$.

Thus by Inverses of Elements Related by Compatible Relation, we obtain the first result:


 * $x \preccurlyeq y \iff y^{-1} \preccurlyeq x^{-1}$

By Reflexive Reduction of Relation Compatible with Group Operation is Compatible, $\prec$ is also compatible with $\circ$.

Thus by again Inverses of Elements Related by Compatible Relation, we obtain the second result:


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