Relation Compatible with Group Operation is Reflexive or Antireflexive

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Theorem

Let $\left({G, \circ}\right)$ be a group.

Let $\mathcal R$ be a relation on $G$ that is compatible with $\circ$.


Then $\mathcal R$ is reflexive or antireflexive.


Proof

Suppose that $\mathcal R$ is not antireflexive.

Then there is some $x \in G$ such that $x \mathrel{\mathcal R} x$.

Let $y \in G$.

Then by the definition of compatibility:

$\left({x \circ \left({x^{-1} \circ y}\right)}\right) \mathrel{\mathcal R} \left({x \circ \left({x^{-1} \circ y}\right)}\right)$

By associativity and the definition of inverse:

$y \mathrel{\mathcal R} y$

Since this holds for arbitrary $y$, $\mathcal R$ is reflexive.

Thus $\mathcal R$ is either reflexive or antireflexive.

$\blacksquare$