Strict Positivity Property induces Total Ordering
Contents
Theorem
Let $\struct {D, +, \times}$ be an integral domain whose zero is $0_D$.
Let $D$ be endowed with a (strict) positivity property $P: D \to \set {\mathrm T, \mathrm F}$.
Then there exists a total ordering $\le$ on $\struct {D, +, \times}$ induced by $P$ which is compatible with the ring structure of $\struct {D, +, \times}$.
Proof
By definition of the strict positivity property:
\((P \, 1)\) | $:$ | 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 $a \in D$. |
Let us define a relation $<$ on $D$ as:
- $\forall a, b \in D: a < b \iff \map P {-a + b}$
Setting $a = 0$:
- $\forall b \in D: 0 < b \iff \map P b$
demonstrating that (strictly) positive elements of $D$ are those which are greater than zero.
From Relation Induced by Strict Positivity Property is Compatible with Addition we have that $<$ is compatible with $+$.
From Relation Induced by Strict Positivity Property is Transitive we have that $<$ is transitive.
From Relation Induced by Strict Positivity Property is Asymmetric and Antireflexive we have that $<$ is asymmetric and antireflexive.
Thus by definition, $<$ is a strict ordering.
Let the relation $\le$ be defined as the reflexive closure of $<$.
From Reflexive Closure of Strict Ordering is Ordering we have that $\le$ is an ordering on $D$.
From Relation Induced by Strict Positivity Property is Trichotomy, and from the Trichotomy Law (Ordering), we have that $\le$ is a total ordering.
Hence the result.
$\blacksquare$
Also see
- Ordered Integral Domain is Totally Ordered Ring: a direct corollary
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order