Reflexive Closure of Relation Compatible with Operation is Compatible
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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 $\circ$.
$\blacksquare$