Ordering Compatible with Group Operation is Strongly Compatible/Corollary/Proof 2

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

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

Let $x, y \in G$.

Then the following equivalences hold:


 * $(\operatorname{OG}2.1):\quad x \preceq y \iff e \preceq y \circ x^{-1}$
 * $(\operatorname{OG}2.2):\quad x \preceq y \iff e \preceq x^{-1} \circ y$


 * $(\operatorname{OG}2.3):\quad x \preceq y \iff x \circ y^{-1} \preceq e$
 * $(\operatorname{OG}2.4):\quad x \preceq y \iff y^{-1} \circ x \preceq e$


 * $(\operatorname{OG}2.1'):\quad x \prec y \iff e \prec y \circ x^{-1}$
 * $(\operatorname{OG}2.2'):\quad x \prec y \iff e \prec x^{-1} \circ y$


 * $(\operatorname{OG}2.3'):\quad x \prec y \iff x \circ y^{-1} \prec e$
 * $(\operatorname{OG}2.4'):\quad x \prec y \iff y^{-1} \circ x \prec e$

Proof
Each result follows from Properties of Ordered Group/OG1. For example, by Properties of Ordered Group/OG1,


 * $x \preceq y \iff x \circ x^{-1} \preceq y \circ x^{-1}$

Since $x \circ x^{-1} = e$:


 * $(\operatorname{OG}2.1):\quad x \preceq y \iff e \preceq y \circ x^{-1}$