Reflexive Reduction of Relation Compatible with Group Operation is Compatible

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

Let $\mathcal R$ be a relation on $S$ which is compatible with $\circ$.

Let $\mathcal R^\ne$ be the reflexive reduction of $\mathcal R$.

Then $\mathcal R^\ne$ is compatible with $\circ$.

Proof
By definition of reflexive reduction, for all $a, b \in S$:


 * $a \mathrel{\mathcal R^\ne} b$ iff $a \mathrel{\mathcal R} b$ but $a \ne b$.

By definition of the diagonal relation $\Delta_S$, $a \ne b$ iff $\left({a, b}\right) \notin \Delta_S$.

Thus, considered as subsets of $S \times S$, we have:


 * $\mathcal R^\ne = \mathcal R \setminus \Delta_S$

By Diagonal Relation is Universally Compatible, $\Delta_S$ is compatible with $\circ$.

Thus by Set Difference of Relations Compatible with Group Operation is Compatible, $\mathcal R^\ne$ is compatible with $\circ$.