# Definition:Ordered Integral Domain/Trichotomy Law

## Definition

Let $\struct {D, +, \times, \le}$ be an ordered integral domain.

Let $P$ be the strict positivity property on $D$

The property:

$\forall a \in D: \map P a \lor \map P {-a} \lor a = 0_D$

is known as the trichotomy law.

That is:

Every element of $D$ is either strictly positive, or strictly negative, or zero.