Definition:Ordered Integral Domain/Definition 1

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

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

An ordered integral domain can be denoted:

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

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

Also see

  • Results about ordered integral domains can be found here.