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 defined by (the strict positivity property) $P$.

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