Properties of Strict Negativity

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 $10$
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