Reflexive Closure of Relation Compatible with Operation is Compatible

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

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

Let $\preceq$ be the reflexive closure of $\prec$.

That is, $\preceq$ is defined as the union of $\prec$ with the diagonal relation for $S$.

Then $\preceq$ 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, $\preceq$ is compatible with $\circ$.

Also See
Reflexive Reduction of Relation Compatible with Group Operation is Compatible