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 an ordering compatible with $\circ$ and $*$.

We recall the definition of the trivial ordering:

Let $a, b \in S$ such that $a \preceq b$.

Since $\preceq$ is compatible with $\circ$ and $*$, we have:

We have that a Boolean ring is an idempotent ring.

Hence we have:

We have shown that:
 * $0 \preceq b - a$

and:
 * $b - a \preceq 0$

By definition of ordering, $\preceq$ is antisymmetric.

This means:
 * $b - a = 0$

and so:
 * $a = b$

Hence $\preceq$ is the trivial ordering.