Properties of Relation Compatible with Group Operation

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Theorem

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

Let $\mathcal R$ be a endorelation on $G$ which is compatible with $\circ$.

Let $x,y,z \in G$.

Then the following hold:

CRG1

$x \mathrel {\mathcal R} y \iff x \circ z \mathrel {\mathcal R} y \circ z$
$x \mathrel {\mathcal R} y \iff z \circ x \mathrel {\mathcal R} z \circ y$


Corollary: CRG2

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


CRG3

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


Corollary: CRG4

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


Relation Compatible with Group Operation is Reflexive or Antireflexive

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.