Category:Ordered Integral Domains

This category contains results about Ordered Integral Domains.
Definitions specific to this category can be found in Definitions/Ordered Integral Domains.

An ordered integral domain is an integral domain $\struct {D, +, \times}$ which has a strict positivity property $P$:

$(\text P 1)$	$:$	Closure under Ring Addition:	$\ds \forall a, b \in D:$	$\ds \map P a \land \map P b \implies \map P {a + b} $
$(\text P 2)$	$:$	Closure under Ring Product:	$\ds \forall a, b \in D:$	$\ds \map P a \land \map P b \implies \map P {a \times b} $
$(\text P 3)$	$:$	Trichotomy Law:	$\ds \forall a \in D:$	$\ds \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} $
		For $\text P 3$, exactly one condition applies for all $a \in D$.

An ordered integral domain is an ordered ring $\struct {D, +, \times, \le}$ which is also an integral domain.

$(\text {OID} 1)$	$:$		$\le$ is compatible with ring addition:	$\ds \forall a, b, c \in D:$		$\ds a \le b $	$\ds \implies $	$\ds \paren {a + c} \le \paren {b + c} $
$(\text {OID} 2)$	$:$		Strict positivity is closed under ring product:	$\ds \forall a, b \in D:$		$\ds 0_D \le a, 0_D \le b $	$\ds \implies $	$\ds 0_D \le a \times b $

An ordered integral domain can be denoted:

$\struct {D, +, \times \le}$

Subcategories

This category has the following 2 subcategories, out of 2 total.

The following 24 pages are in this category, out of 24 total.

\((\text P 1)\)	$:$	Closure under Ring Addition:	\(\ds \forall a, b \in D:\)	\(\ds \map P a \land \map P b \implies \map P {a + b} \)
\((\text P 2)\)	$:$	Closure under Ring Product:	\(\ds \forall a, b \in D:\)	\(\ds \map P a \land \map P b \implies \map P {a \times b} \)
\((\text P 3)\)	$:$	Trichotomy Law:	\(\ds \forall a \in D:\)	\(\ds \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} \)
		For $\text P 3$, exactly one condition applies for all $a \in D$.

\((\text {OID} 1)\)	$:$		$\le$ is compatible with ring addition:	\(\ds \forall a, b, c \in D:\)		\(\ds a \le b \)	\(\ds \implies \)	\(\ds \paren {a + c} \le \paren {b + c} \)
\((\text {OID} 2)\)	$:$		Strict positivity is closed under ring product:	\(\ds \forall a, b \in D:\)		\(\ds 0_D \le a, 0_D \le b \)	\(\ds \implies \)	\(\ds 0_D \le a \times b \)