Definition:Ordered Integral Domain

Definition
An ordered integral domain $\left({D, +, \times}\right)$ is an integral domain $\left({D, +, \times}\right)$ with a property $P$ such that:


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


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


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

For condition $(3)$, exactly one of the conditions applies for every element of $D$.

The property $P$ 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.

Also see
From Positivity Property induces Total Ordering it can be seen that this definition is equivalent to that of a totally ordered ring.