Inverse of Relation Compatible with Operation is Compatible

Theorem

Let $\struct {S, \circ}$ be a closed algebraic structure.

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

Let $\QQ$ be the inverse relation of $\RR$.

Then $\QQ$ is compatible with $\circ$.

Proof

Let $x, y, z \in S$.

Suppose that $x \mathrel \QQ y$.

Then by the definition of $\QQ$:

$y \mathrel \RR x$.

Since $\RR$ is compatible with $\circ$:

$\paren {y \circ z} \mathrel \RR \paren {x \circ z}$

and

$\paren {z \circ y} \mathrel \RR \paren {z \circ x}$

Thus by the definition of $\QQ$:

$\paren {x \circ z} \mathrel \QQ \paren {y \circ z}$

and

$\paren {z \circ x} \mathrel \RR \paren {z \circ y}$

Thus $\QQ$ is compatible with $\circ$.

$\blacksquare$