Definition:Positivity Property
Definition
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $P \subseteq R$ such that:
\((\text P 1)\) | $:$ | \(\ds P + P \subseteq P \) | |||||||
\((\text P 2)\) | $:$ | \(\ds P \cap \paren {-P} = \set {0_R} \) | |||||||
\((\text P 3)\) | $:$ | \(\ds P \circ P \subseteq P \) |
Let $\PP: R \to \set {\T, \F}$ be the propositional function defined as:
- $\forall x \in D: \map \PP x \iff x \in P$
Then $\PP$ 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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.12$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields