Reflexive Closure of Relation Compatible with Operation is Compatible

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Theorem

Let $\left({S, \circ}\right)$ be a magma.

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


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

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

Then $\mathcal R^=$ 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, $\mathcal R^=$ is compatible with $\circ$.

$\blacksquare$


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