User:Dfeuer/Cone Condition Equivalent to Reflexivity

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

Let $C$ be a cone compatible with $\circ$.

Let $\mathcal R$ be the compatible relation on $G$ induced by $C$.

Then the following are equivalent:
 * $\mathcal R$ is reflexive.
 * $e \in C$
 * $e \in C^{-1}$

Proof
Suppose that $\mathcal R$ is reflexive.

Then $e \mathrel{\mathcal R} e$.

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

Thus $e^{-1} \in C^{-1}$.

Since $e^{-1} = e$, $e \in C^{-1}$.

Suppose instead that $e \in C$.

Let $x \in R$.

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

Thus $x \mathrel{\mathcal R} x$.

Suppose instead that $e \in C^-1$.

Then $e = e^{-1} \in C$.