[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?