Definition:Strict Negativity Property
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Definition
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$.
The strict negativity property $N$ is defined as:
- $\forall a \in D: \map N a \iff \map P {-a}$
This is compatible with the trichotomy law:
- $\forall a \in D: \map P a \lor \map P {-a} \lor a = 0_D$
which can therefore be rewritten:
- $\forall a \in D: \map P a \lor \map N a \lor a = 0_D$
or even:
- $\forall a \in D: \map N a \lor \map N {-a} \lor a = 0_D$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order