# Definition:Positivity Property

## Definition

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $P \subseteq R$ such that:

 $(P \, 1)$ $:$ $\displaystyle P + P \subseteq P$ $(P \, 2)$ $:$ $\displaystyle P \cap \paren {-P} = \set {0_R}$ $(P \, 3)$ $:$ $\displaystyle P \circ P \subseteq P$

Let $\mathcal P: R \to \set {\mathrm T, \mathrm F}$ be the propositional function defined as:

$\forall x \in D: \mathcal P x \iff x \in P$

Then $\mathcal P$ is the positivity property on $\struct {R, +, \circ}$.

## Also defined as

The name positivity property is also defined to be the similar propositional function, usually defined on an integral domain $\struct {D, +, \times}$ which does not include zero in its fiber of truth.

Because $\struct {R, +, \circ}$ may have (proper) zero divisors, such a propositional function may not be closed under $\circ$.

Hence it is the intention on $\mathsf{Pr} \infty \mathsf{fWiki}$ to refer consistently to that propositional function as the strict positivity property, and to reserve positivity property for this one.