Properties of Relation Compatible with Group Operation/CRG2

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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:

$(1): \quad x \mathrel {\mathcal R} y \iff e \mathrel {\mathcal R} y \circ x^{-1}$
$(2): \quad x \mathrel {\mathcal R} y \iff e \mathrel {\mathcal R} x^{-1} \circ y$
$(3): \quad x \mathrel {\mathcal R} y \iff x \circ y^{-1} \mathrel {\mathcal R} e$
$(4): \quad x \mathrel {\mathcal R} y \iff y^{-1} \circ x \mathrel {\mathcal R} e$


Proof

We can apply CRG1 to $x$, $y$, and $x^{-1}$ to obtain the following equivalences:

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

Applying CRG1 to $x$, $y$, and $y^{-1}$, on the other hand, yields:

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


By the definition of inverses:

$x \circ x^{-1} = x^{-1} \circ x = y \circ y^{-1} = y^{-1} \circ y = e$

Making these substitutions proves the theorem.

$\blacksquare$