Relation Induced by Strict Positivity Property is Compatible with Multiplication

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:

$\forall x, y, z \in D: x < y, \map P z \implies \paren {z \times x} < \paren {z \times y}$

$\forall x, y, z \in D: x < y, \map P z \implies \paren {x \times z} < \paren {y \times z}$

If $x < y$ then $\map P {-x + y}$.

Hence:

$\ds $

$\ds \map P {-x + y}, \, \map P z$

$\ds $

$\leadsto$

$\ds \map P {z \times \paren {-x + y} }$

$\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.

$\blacksquare$

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order: Theorem $9: \ \text{O} 2$