Ordering Compatible with Group Operation is Strongly Compatible/Corollary

Theorem
Let $\left({G, \circ, \le}\right)$ be an ordered group with identity $e$.

Let $x, y \in G$.

Then the following equivalences hold:


 * $x \le y \iff e \le y \circ x^{-1}$ (1)
 * $x \le y \iff e \le x^{-1} \circ y$ (2)


 * $x \le y \iff x \circ y^{-1} \le e$ (3)
 * $x \le y \iff y^{-1} \circ x \le e$ (4)


 * $x < y \iff e < y \circ x^{-1}$ (1')
 * $x < y \iff e < x^{-1} \circ y$ (2')


 * $x < y \iff x \circ y^{-1} < e$ (3')
 * $x < y \iff y^{-1} \circ x < e$ (4')

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

Thus by User:Dfeuer/CRG2, we obtain the first four results.

By Reflexive Reduction of Relation Compatible with Group Operation is Compatible, $< \mathop = \le^\ne$ is compatible with $\circ$.

Thus by User:Dfeuer/CRG2, we obtain the remaining results.