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

Theorem
Let $\struct {G, \circ}$ be a group with identity $e$.

Let $\RR$ be a transitive relation compatible with $\circ$.

Let $C = \set {x \in G: e \mathrel \RR x}$.

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

Existence
Suppose $a \mathrel \RR b$.

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

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

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

So $a \mathrel \RR b$.

Thus $C$ induces $\RR$.

Uniquelness
Suppose $D$ induces $\RR$.

Let $x \in C$.

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

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

Let $x \in D$.

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

So $e \mathrel \RR x$, so $x \in C$.