# 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:

The trivial ordering is an ordering $\RR$ defined on a set $S$ by:

$\forall a, b \in S: a \mathrel \RR b \iff a = b$

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

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

 $\ds a$ $\preceq$ $\ds b$ $\ds \leadsto \ \$ $\ds 0$ $\preceq$ $\ds b - a$

We have a fortiori that a Boolean ring is an idempotent ring.

Hence we have:

 $\ds a$ $\preceq$ $\ds b$ $\ds \leadsto \ \$ $\ds a - b$ $\preceq$ $\ds 0$ $\ds \leadsto \ \$ $\ds b - a$ $\preceq$ $\ds 0$ Idempotent Ring has Characteristic Two: Corollary

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.

$\blacksquare$