Definition:Positivity Property

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

Let $P \subseteq R$ such that:

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 to refer consistently to that propositional function as the strict positivity property, and to reserve positivity property for this one.

Also see

 * Definition:Ordered Ring