Trichotomy Law (Ordering)

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Then $\preceq$ is a total ordering if and only if:

$\forall a, b \in S: \paren {a \prec b} \lor \paren {a = b} \lor \paren {a \succ b}$

That is, every element either strictly precedes, is the same as, or strictly succeeds, every other element.

In other words, if and only if $\prec$ is a trichotomy.

$\ds \forall a, b \in S: \, $

$\ds $

$\ds a \preceq b \lor b \preceq a$

Definition of Total Ordering

$\ds \leadstoandfrom \ \ $

$\ds \forall a, b \in S: \, $

$\ds $

$\ds a \preceq b \lor a \succeq b$

Definition of Dual Ordering

$\ds \leadstoandfrom \ \ $

$\ds \forall a, b \in S: \, $

$\ds $

$\ds \paren {a = b \lor a \prec b} \lor \paren {a = b \lor a \succ b}$

$\ds \leadstoandfrom \ \ $

$\ds \forall a, b \in S: \, $

$\ds $

$\ds a \prec b \lor a = b \lor a \succ b$

$\blacksquare$

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

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order