# 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.

$\blacksquare$

## Sources

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