Relation Induced by Strict Positivity Property is Transitive

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:

$\forall a, b \in D: a < b \iff \map P {-a + b}$

Let $a < b$ and $b < c$.

Thus:

$\ds $

$\ds a < b, b < c$

$\ds $

$\leadsto$

$\ds \map P {-a + b}, \map P {-b + c}$

Definition of $<$

$\ds $

$\leadsto$

$\ds \map P {\paren {-a + b} + \paren {-b + c} }$

$\ds $

$\leadsto$

$\ds \map P {-a + c}$

Properties of $+$ in $D$

$\ds $

$\leadsto$

$\ds a < c$

Definition of $<$

And so $<$ is seen to be transitive.

$\blacksquare$

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order: Theorem $8$