Order axioms - Knowlpedia

Order axioms: Let $F$ be a field (satisfying the field axioms) and let $\le$ be a relation on $F$ . The following are required for all $a,b,c\in F$ :

(Reflexivity) $a\le a$ .
(Antisymmetry) if $a\le b$ and $b\le a$ , then $a=b$ .
(Transitivity) if $a\le b$ and $b\le c$ , then $a\le c$ .
(Totality) for any $a,b$ , either $a\le b$ or $b\le a$ .
(Compatibility with addition) if $a\le b$ , then $a+c\le b+c$ .
(Compatibility with multiplication) if $0\le a$ and $0\le b$ , then $0\le ab$ .

Remarks

A field equipped with such an order is an ordered field; the real numbers form the standard example. The order interacts with the absolute value and underlies definitions of intervals, bounds, and $\varepsilon$ – $\delta$ limits.