Ordering Compatible with Group Operation is Strongly Compatible

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

Let $x, y, z \in G$.

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 Properties of Relation Compatible with Group Operation/CRG1, we obtain the first two results:


 * $(\operatorname{OG}1.1):\quad x \preceq y \iff x \circ z \preceq y \circ z$
 * $(\operatorname{OG}1.2):\quad x \preceq y \iff z \circ x \preceq z \circ y$

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

Thus by again Properties of Relation Compatible with Group Operation/CRG1, we obtain the remaining results:


 * $(\operatorname{OG}1.1'):\quad x \prec y \iff x \circ z \prec y \circ z$
 * $(\operatorname{OG}1.2'):\quad x \prec y \iff z \circ x \prec z \circ y$

and so the theorem is established.