Relation Induced by Strict Positivity Property is Compatible with Multiplication
Jump to navigation
Jump to search
Theorem
Let $\struct {D, +, \times}$ be an ordered integral domain where $P$ is the (strict) positivity property.
Let the relation $<$ be defined on $D$ as:
- $\forall a, b \in D: a < b \iff \map P {-a + b}$
Then $<$ is compatible with $\times$ in the following sense:
- $\forall x, y, z \in D: x < y, \map P z \implies \paren {z \times x} < \paren {z \times y}$
- $\forall x, y, z \in D: x < y, \map P z \implies \paren {x \times z} < \paren {y \times z}$
Proof
If $x < y$ then $\map P {-x + y}$.
Hence:
\(\ds \) | \(\) | \(\ds \map P {-x + y}, \, \map P z\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \map P {z \times \paren {-x + y} }\) | Definition of Strict Positivity Property | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \map P {-z \times x + z \times y}\) | Distributivity of $\times$ over $+$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds z \times x < z \times y\) |
The other result follows from the fact that $\times$ is commutative in an integral domain.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order: Theorem $9: \ \text{O} 2$