Definition:Ordered Integral Domain/Trichotomy Law
< Definition:Ordered Integral Domain(Redirected from Definition:Trichotomy Law (Integral Domain))
Jump to navigation
Jump to search
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