User:Dfeuer/Cone Condition Equivalent to Symmetry

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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 symmetric.
  • $C = C^{-1}$


Proof