Suppose I have a function $f : 2^\Omega \to 2^{\Omega \times \Omega}$ such that: $\forall S \subseteq \Omega$, $f(S)$ is a partial order on $S$.
Is there a name for such a function? If not, how would you name it?
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I'd call it a "partial-ordering function," or some such. I'm not aware of any standard nomenclature.
Incidentally, you can explicitly construct such functions for any $\Omega.$ Let $R$ be some partial-ordering relation on $\Omega$, and for each $S\subseteq\Omega,$ let $f(S)=R\cap(S\times S)$. Since a partial order on $\Omega$ automatically induces compatible partial orders on all subsets of $\Omega,$ it seems to me that arbitrary functions of the sort you describe would be of limited utility.
Cameron Buie
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That's exactly what I'm interested in. I'm trying to figure out if a function that induces incompatible partial orders may have some utility and how should one deal with such functions. – Petro Aldarri Oct 25 '13 at 19:28