Definition:Total Ordering Induced by Strict Positivity Property
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Definition
Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$.
Let $P: D \to \set {\T, \F}$ denote the strict positivity property:
\((\text P 1)\) | $:$ | 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 $a \in D$. |
Then the total ordering $\le$ compatible with the ring structure of $D$ is called the (total) ordering induced by (the strict positivity property) $P$.
Also known as
The (total) ordering induced by (the strict positivity property) $P$ is also seen as (total) ordering defined by (the strict positivity property) $P$.
The strict positivity property is generally known as the positivity property, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ we place emphasis on the strictness.
Also see
This ordering is shown to exist by Strict Positivity Property induces Total Ordering.
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields