Definition:Total Ordering Induced by Strict Positivity Property

Definition
Let $\struct {D, +, \times, \le}$ be an ordered integral 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:

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 we place emphasis on the strictness.

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