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, this 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