Reflexive Closure of Relation Compatible with Operation is Compatible

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

Let $\prec$ be a transitive, antisymmetric Definition:Endorelation on $S$ which is compatible with $\circ$.

Then $\preceq = \prec \cup \Delta_S$ is compatible with $\circ$,

where $\Delta_S$ is the diagonal relation for $S$.

Also See
Transitive Antisymmetric Relation Compatible with Group Induces Compatible Strict Ordering