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 $23.34$