User:Dfeuer/Cone Condition Equivalent to Irreflexivity

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 irreflexive.
• $\mathcal R$ is not reflexive.
• $e \notin C$
• $e \notin C^{-1}$

Proof

Relation Compatible with Group is Reflexive or Irreflexive and Cone Condition Equivalent to Reflexivity