Definition:Ordered Integral Domain

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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:

$(1): \quad$ 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.


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

  • 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