Definition:Ordered Integral Domain/Definition 1
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Definition
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 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.
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order