Define for posets with order $latex \sqsubseteq$:

- $latex \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$;
- $latex \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \}$.

Note that the above is a generalization of monotone Galois connections (with $latex \max$ and $latex \min$ replaced with suprema and infima).

Then we get the following diagram (see this PDF file for a proof):

It is yet unknown what will happen if we keep apply $latex \Phi_{\ast}$ and/or $latex \Phi^{\ast}$ to the node “other”. Will this lead to a finite or infinite set?

The diagram was with an error. I have edited the post.