Trivial Ordering Compatibility in Boolean Ring

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Let $\struct {S, +, \circ}$ be a Boolean ring.

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


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:

$0 \preceq b - a$


$b - a \preceq 0$

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

This means:

$b - a = 0$

and so:

$a = b$

Hence $\preceq$ is the trivial ordering.