Ordering Compatible with Group Operation is Strongly Compatible

Theorem
Let $\struct {G, \circ, \preceq}$ 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 Relation Compatible with Group Operation is Strongly Compatible, we obtain the first two results:

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

Thus again by Relation Compatible with Group Operation is Strongly Compatible, we obtain the remaining results:

and so the theorem is established.