# Definition:Positivity Property

## Definition

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

Let $P \subseteq R$ such that:

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

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

$\forall x \in D: \map \PP x \iff x \in P$

Then $\PP$ 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.