Definition:Total Ordering Induced by Strict Positivity Property

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Definition

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


Let $P: D \to \set {\mathrm T, \mathrm F}$ 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.


Sources