Definition:Strict Positivity Property

Definition
Let $\left({D, +, \times, \le}\right)$ be an ordered integral domain, where $\le$ is the ordering induced by the property $P$:


 * $(1): \quad \forall a, b \in D: P \left({a}\right) \land P \left({b}\right) \implies P \left({a + b}\right)$


 * $(2): \quad \forall a, b \in D: P \left({a}\right) \land P \left({b}\right) \implies P \left({a \times b}\right)$


 * $(3): \quad \forall a \in D: P \left({a}\right) \lor P \left({-a}\right) \lor a = 0_D$

The property $P$ is called the positivity property.

As its name implies, it is identified with the property of being positive.

Hence the above conditions can be written in natural language as:


 * $(1): \quad$ The sum of any two positive elements is also positive.


 * $(2): \quad$ The product of any two positive elements is also positive.


 * $(3): \quad$ Every element is either positive, or negative, or zero.