Definition:Positivity Property

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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.


Also see


Sources