[Jesus Lopez wrote](https://forum.azimuthproject.org/discussion/comment/16333/#Comment_16333):

> Galois connection between the posets of partions of two sets \\(S,T\\) is built, given any function \\(g:S \to T\\). Can this construction be generalized for arbitrary relations \\(G \subseteq S \times T\\), so one still attains a Galois connection as before?

That's a great question, but I don't instantly know the answer: I'd have to think about it for a while. For starters, I bet there are a couple of ways you could take a relation \\(G \subseteq S \times T\\) and a partition of \\(S\\) and get a partition of \\(T\\). I'd have to start by investigating the properties of these ways, though surely it's been done before, so if I were feeling lazy I'd just do a literature search or ask my pals.

Did you have a specific way in mind?

> Galois connection between the posets of partions of two sets \\(S,T\\) is built, given any function \\(g:S \to T\\). Can this construction be generalized for arbitrary relations \\(G \subseteq S \times T\\), so one still attains a Galois connection as before?

That's a great question, but I don't instantly know the answer: I'd have to think about it for a while. For starters, I bet there are a couple of ways you could take a relation \\(G \subseteq S \times T\\) and a partition of \\(S\\) and get a partition of \\(T\\). I'd have to start by investigating the properties of these ways, though surely it's been done before, so if I were feeling lazy I'd just do a literature search or ask my pals.

Did you have a specific way in mind?