Relation Induced by Strict Positivity Property is Transitive

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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 a transitive relation.


Proof

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

Thus:

\(\displaystyle \) \(\) \(\displaystyle a < b, b < c\)
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map P {-a + b}, \map P {-b + c}\) Definition of $<$
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map P {\paren {-a + b} + \paren {-b + c} }\) Definition of Strict Positivity Property
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map P {-a + c}\) Properties of $+$ in $D$
\(\displaystyle \) \(\leadsto\) \(\displaystyle a < c\) Definition of $<$

And so $<$ is seen to be transitive.

$\blacksquare$


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