Reflexive Closure of Relation Compatible with Operation is Compatible

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

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

Let $\RR^=$ be the reflexive closure of $\prec$.

That is, $\RR^=$ is defined as the union of $\RR$ with the diagonal relation for $S$.

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

Proof
By Diagonal Relation is Universally Compatible, the diagonal relation is compatible with $\circ$.

Then by Union of Relations Compatible with Operation is Compatible, $\RR^=$ is compatible with $\circ$.

Also see

 * Reflexive Reduction of Relation Compatible with Group Operation is Compatible