Let $R$ be a ring and $\leq$ be an order. $R$ is ordered by $\leq$ if:
(Addition preserves order) For all $a$, $b$, $x \in R$,
\[a \leq b \enspace\rimp\enspace a + x \leq b + x.\](Multiplication preserves non-negativity) For all $a$, $b \in R$,
\[0 \leq a \,\land\, 0 \leq b \enspace\rimp\enspace 0 \leq ab.\]