= Pointwise order on real polynomials
{title2=$p\preceq q\iff p(x)\le q(x)\text{ for every }x\in\mathbb R$}
This is a <partial order>: the real inequalities supply <reflexivity> and <transitivity>, while inequalities in both directions give equality of the <polynomials> as functions. It is not a <total order>, since the <polynomials> $x$ and $0$ are incomparable on all of $\mathbb R$. Restricting the test domain can change the relation; testing only at finitely many points need not distinguish different <polynomials>.
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