Reflexive Reduction of Relation Compatible with Group Operation is Compatible

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

Let $<$ be a transitive, antisymmetric relation on $S$ which is compatible with $\circ$.

Then $\lneqq = <\setminus \Delta_S$ is compatible with $\circ$.

Proof
Let $\Delta$ be the diagonal relation on $S$ and let $\Delta^C$ be its complement in $S \times S$.

By Diagonal Complement Relation Compatible with Group, $\Delta^C$ is compatible with $\circ$.

Since $\lneqq = \Delta^C \cap <$ and Intersection of Relations Compatible with Operation is Compatible, $\lneqq$ is compatible with $\circ$.

$\lneqq$ is a strict ordering.