User:Dfeuer/Cone Condition Equivalent to Asymmetry

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 asymmetric.
 * $C \cap C^{-1} = \varnothing$

Proof
Follows from Cone Condition Equivalent to Irreflexivity and Cone Condition Equivalent to Antisymmetry.