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 $a$, $b\in R$,
\[0\leq a \,\land\, 0\leq b \enspace\rimp\enspace 0\leq ab.\]