Order axioms

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 $0\le ab$.

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.