Properties of Relation Compatible with Group Operation/CRG4

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Theorem

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

Let $\mathcal R$ be a relation compatible with $\circ$.

Let $x, y \in G$.


Then the following equivalences hold:

$x \mathrel{\mathcal R} e \iff e \mathrel{\mathcal R} x^{-1}$
$e \mathrel{\mathcal R} x \iff x^{-1} \mathrel{\mathcal R} e$


Proof

Applying User:Dfeuer/CRG2$(1)$ to $x$ and $e$ gives:

$x \mathrel{\mathcal R} e \iff e \mathrel{\mathcal R} e \circ x^{-1}$

Applying User:Dfeuer/CRG2$(3)$ to $e$ and $x$ gives:

$e \mathrel{\mathcal R} x \iff e \circ x^{-1} \mathrel{\mathcal R} e$


Since $e \circ x^{-1} = x^{-1} \circ e = x^{-1}$ for all $x \in G$, the theorem holds.

$\blacksquare$