Category:Ordered Integral Domains

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

Definition 1

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 $a \in D$.      

Definition 2

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

That is, it is an integral domain with an ordering $\le$ compatible with the ring structure of $\struct {D, +, \times}$:

\((\text {OID} 1)\)   $:$   $\le$ 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:

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

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


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

Pages in category "Ordered Integral Domains"

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