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:

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 to refer consistently to the propositional function as defined on this page as the strict positivity property.

Also see

 * Definition:Ordered Integral Domain