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.

Also known as

The Trichotomy Law can also be seen referred to as the trichotomy principle.

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order