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 $\prec$)?

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    $\begingroup$ I can't understand your question. It sounds like you are looking for a complete partial order for some set, but (1) I can't tell what the domain/set is, and (2) I can't tell what specific requirements you want it to satisfy. What do you mean by "partial functions allowing for different outputs"? Try breaking your sentences down into smaller pieces and defining your concepts precisely. For instance, define a set $S$ that is the set of partial functions you want to consider, then ask for a complete partial order on $S$ such that (fill in this part). $\endgroup$
    – D.W.
    Apr 5 '15 at 0:00

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