Strict Negativity is equivalent to Strict Positivity of Negative
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Theorem
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$.
Let $N$ be the (strict) negativity property on $D$:
- $\forall a \in D: \map N a \iff \map P {-a}$
Then for all $a \in D$:
- $\map P a \iff \map N {-a}$
Proof
\(\ds \map P a\) | \(\leadstoandfrom\) | \(\ds \map P {-\paren {-a} }\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \map N {-a}\) | Definition of Strict Negativity Property |
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Ordered and Well-Ordered Integral Domains: $\S 7$. Order: Theorem $10 \ \text {(ii)}$