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.