Conditions for Ordering in Ordered Group to be Directed

Theorem
Let $\struct {G, \odot, \preccurlyeq}$ be an ordered group whose identity element is $e$.

Then:
 * $\preccurlyeq$ is a directed ordering


 * for every $x \in G$ there exist $y, z \in G$ such that $e \preccurlyeq y$, $e \preccurlyeq z$ and $x = y \odot z^{-1}$.

Sufficient Condition
Let $\preccurlyeq$ be a directed ordering.

By definition of directed ordering:
 * $\forall x, z \in G: \exists y \in G: x \preccurlyeq y$ and $z \preccurlyeq y$

Let $x \in G$ be arbitrary.

Let $e = \paren {y \odot x^{-1} }^{-1} \odot z$.

Then:

Then from:

we get:

and we see that:


 * $\exists y, z \in G: e \preccurlyeq y, e \preccurlyeq z, x = y \odot z^{-1}$

Necessary Condition
Let $\preccurlyeq$ be such that:
 * for every $x \in G$ there exist $y, z \in G$ such that $e \preccurlyeq y$, $e \preccurlyeq z$ and $x = y \odot z^{-1}$.

Let $x, z \in G$ be arbitrary.

Let $y = x \odot z$.

We have:

Then we have: