Trivial Ordering Compatibility in Boolean Ring

Theorem

Let $\struct {S, +, \circ}$ 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 $*$.