Reflexive Reduction of Relation Compatible with Group Operation is Compatible
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 1
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 $\circ$.
$\blacksquare$
Proof 2
From the Cancellation Laws, $\circ$ is a cancellable operation.
The result then follows directly from Reflexive Reduction of Relation Compatible with Cancellable Operation is Compatible.
$\blacksquare$