Definition:Strict Positivity Property
Contents
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:
\((P \, 1)\) | $:$ | Closure under Ring Addition: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle \map P a \land \map P b \implies \map P {a + b} \) | ||||
\((P \, 2)\) | $:$ | Closure under Ring Product: | \(\displaystyle \forall a, b \in D:\) | \(\displaystyle \map P a \land \map P b \implies \map P {a \times b} \) | ||||
\((P \, 3)\) | $:$ | Trichotomy Law: | \(\displaystyle \forall a \in D:\) | \(\displaystyle \map P a \lor \map P {-a} \lor a = 0_D \) | ||||
For $P \, 3$, 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, 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.1$: The integers
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields