# Definition:Ordered Integral Domain

## Definition

An **ordered integral domain** 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 $D$.

An ordered integral domain can be denoted:

- $\left({D, +, \times \le}\right)$

where $\le$ is the ordering induced by the positivity property.

### Positivity

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:

### Trichotomy Law

The property:

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

is known as the **trichotomy law**.

## Also see

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

- Trichotomy Law (in the context of an ordering). Note that the two statements of the
**trichotomy law**are ultimately equivalent, but the one given on this page is ultimately more general and fundamental.

- Results about
**ordered integral domains**can be found here.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 2.7$