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:
\((\text P 1)\) | $:$ | Closure under Ring Addition: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle \map P a \land \map P b \implies \map P {a + b} \) | ||||
\((\text P 2)\) | $:$ | Closure under Ring Product: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle \map P a \land \map P b \implies \map P {a \times b} \) | ||||
\((\text P 3)\) | $:$ | Trichotomy Law: | \(\displaystyle \forall a \in D:\) | \(\displaystyle \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D} \) | ||||
For $\text P 3$, exactly one condition applies for all $a \in D$. |
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 $\mathsf{Pr} \infty \mathsf{fWiki}$ to refer consistently to the propositional function as defined on this page as the strict positivity property.
Also see
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers