User:Dfeuer/Transitive Relation Compatible with Group Operation Induced by Unique Cone

Theorem
Let $(G,\circ)$ be a group with identity $e$.

Let $\mathrel{\mathcal R}$ be a transitive relation compatible with $\circ$.

Let $C = \{x \in G: e \mathrel{\mathcal R} x \}$.

Then $C$ induces $\mathcal R$ and is the only compatible cone to do so.

induces
Suppose $a \mathrel{\mathcal R} b$.

Then $a \circ a^{-1} \mathrel{\mathcal R} b \circ a^{-1}$, so
 * $e \mathrel{\mathcal R} b \circ a^{-1}$

so $b \circ a^{-1} \in C$.

If $b \circ a^{-1} \in C$, then
 * $e \mathrel{\mathcal R} b \circ a^{-1}$

So $a \mathrel{\mathcal R} b$.

Thus $C$ induces $\mathcal R$.

Uniquely
Suppose $D$ induces $\mathcal R$.

Let $x \in C$.

Then $x \circ e^{-1} = x$, so $e \mathrel{\mathcal R} x$.

Thus $x=x \circ e^{-1} \in D$.

Let $x \in D$.

Then $x \circ e^{-1} = x \in D$.

So $e \mathrel{\mathcal R} x$, so $x \in C$.