Since a partial function can be seen as a set of tuples, there is a trivial CPO defined by the subset relation on partial functions of the same (co-)domain. However, this is not really useful. What I'd need is a CPO over partial functions allowing for different outputs. I.e. two functions are ordered $f < g$ when, for all $f(x)\prec g(x)$ for some CPO $\prec$ on the codomain. It seems intuitive that this is also a CPO, but is it really? Is there a theorem out there somewhere proving it (given some properties of $\prec$)?