Trivial Ordering Compatibility in Boolean Ring

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

Then the trivial ordering is the only ordering on $S$ compatible with both its operations.

Proof
That the trivial ordering is compatible with $\circ$ and $*$ follows from Trivial Ordering is Universally Compatible.

Conversely, suppose that $\preceq$ is a ordering compatible with $\circ$ and $*$.