# Definition:Strict Positivity Property

## Definition

Let $\struct {D, +, \times}$ be an integral domain whose zero is $0_D$.

Let $\struct {D, +, \times}$ be such that a propositional function $P: D \to \set {\mathrm T, \mathrm F}$ can be applied to a subset of $D$ as follows:

 $(P \, 1)$ $:$ Closure under Ring Addition: $\displaystyle \forall a, b \in D:$ $\displaystyle \map P a \land \map P b \implies \map P {a + b}$ $(P \, 2)$ $:$ Closure under Ring Product: $\displaystyle \forall a, b \in D:$ $\displaystyle \map P a \land \map P b \implies \map P {a \times b}$ $(P \, 3)$ $:$ Trichotomy Law: $\displaystyle \forall a \in D:$ $\displaystyle \map P a \lor \map P {-a} \lor a = 0_D$ For $P \, 3$, exactly one condition applies for all $a \in D$.

The propositional function $P$ as defined above is called the strict positivity property.

## Also known as

This is usually known in the literature as the positivity property.

However, the term positivity property is also used to define a similar propositional function, usually defined on a general ring $\struct {R, +, \circ}$ which includes zero in its fiber of truth.

Because $\struct {R, +, \circ}$ may have (proper) zero divisors, $P$ as defined here may not be closed under $\circ$.

Hence it is the intention on $\mathsf{Pr} \infty \mathsf{fWiki}$ to refer consistently to the propositional function as defined on this page as the strict positivity property.