Properties of Strict Negativity
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Theorem
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$.
Let $\le$ be the total ordering induced by $P$, and let $<$ be its strict total ordering counterpart.
Let $N$ be the (strict) negativity property on $D$:
- $\forall a \in D: \map N a \iff \map P {-a}$
Then the following properties apply for all $a, b \in D$:
Strict Negativity is equivalent to Strictly Preceding Zero
- $\map N a \iff a < 0$
Strict Negativity is equivalent to Strict Positivity of Negative
- $\map P a \iff \map N {-a}$
Sum of Strictly Negative Elements is Strictly Negative
- $\map N a, \map N b \implies \map N {a + b}$
Product of Two Strictly Negative Elements is Strictly Positive
- $\map N a, \map N b \implies \map P {a \times b}$
Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative
- $\map N a, \map P b \implies \map N {a \times b}$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): $\S 2.7$: Theorem $10$