Definition:Ordered Integral Domain
Definition
Definition 1
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 $a \in D$. |
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 $\struct {D, +, \times}$:
\((OID \, 1)\) | $:$ | $\le$ is compatible with ring addition: | \(\displaystyle \forall a, b, c \in D:\) | \(\displaystyle a \le b \) | \(\displaystyle \implies \) | \(\displaystyle \paren {a + c} \le \paren {b + c} \) | ||
\((OID \, 2)\) | $:$ | Strict positivity is closed under ring product: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle 0_D \le a, 0_D \le b \) | \(\displaystyle \implies \) | \(\displaystyle 0_D \le a \times b \) |
An ordered integral domain can be denoted:
- $\struct {D, +, \times \le}$
where $\le$ is the total ordering induced by the strict positivity property.
Also see
- Trichotomy Law (in the context of a general ordering)
- Results about ordered integral domains can be found here.