Is polynomial ring a lattice?

Question

My prof says it's not. But I can't find a polynomial pair of $f,g$ such that $max(f,g)$ or $min(f,g)$ is not in $R[x]$. Define uniform order: $f\leq g$,if for all $x, f\leq g $.

Answer 1

Note that the ordering you give forms a lattice over the set of functions continuous in an interval, so the problem must be that a supremum isn't a polynomial.

As git suggests in a comment above, you can take $f(x) = x, g(x) = - x$. I claim that not only is there no supremum of these two functions, but in fact there is no polynomial greater than both of them under your ordering. Can you see why?