# 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.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Exercise $23.34$