# Reflexive Reduction of Relation Compatible with Group Operation is Compatible

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## Theorem

Let $\struct {S, \circ}$ be a group.

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

Let $\RR^\ne$ be the reflexive reduction of $\RR$.

Then $\RR^\ne$ is compatible with $\circ$.

## Proof

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

- $a \mathrel {\RR^\ne} b$ if and only if $a \mathrel \RR b$ but $a \ne b$.

By definition of the diagonal relation $\Delta_S$:

- $a \ne b$ if and only if $\tuple {a, b} \notin \Delta_S$

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

- $\RR^\ne = \RR \setminus \Delta_S$

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

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

$\blacksquare$