# Definition:Total Ordering Induced by Strict Positivity Property

## Definition

Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$.

Let $P: D \to \set {\T, \F}$ denote the strict positivity property:

 $(\text P 1)$ $:$ Closure under Ring Addition: $\ds \forall a, b \in D:$ $\ds \map P a \land \map P b \implies \map P {a + b}$ $(\text P 2)$ $:$ Closure under Ring Product: $\ds \forall a, b \in D:$ $\ds \map P a \land \map P b \implies \map P {a \times b}$ $(\text P 3)$ $:$ Trichotomy Law: $\ds \forall a \in D:$ $\ds \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D}$ For $\text P 3$, exactly one condition applies for all $a \in D$.

Then the total ordering $\le$ compatible with the ring structure of $D$ is called the (total) ordering induced by (the strict positivity property) $P$.

## Also known as

The (total) ordering induced by (the strict positivity property) $P$ is also seen as (total) ordering defined by (the strict positivity property) $P$.

The strict positivity property is generally known as the positivity property, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ we place emphasis on the strictness.

## Also see

This ordering is shown to exist by Strict Positivity Property induces Total Ordering.