Definition:Strict Negativity Property

Definition
Let $\left({D, +, \times}\right)$ be an ordered integral domain, whose positivity property is denoted $P$.

The negativity property $N$ is defined as:
 * $\forall a \in D: N \left({a}\right) \iff P \left({-a}\right)$

This is compatible with the trichotomy law:


 * $\forall a \in D: P \left({a}\right) \lor P \left({-a}\right) \lor a = 0_D$

which can therefore be rewritten:
 * $\forall a \in D: P \left({a}\right) \lor N \left({a}\right) \lor a = 0_D$