![](/files/happy5.png)
Define for posets with order :
;
.
Note that the above is a generalization of monotone Galois connections (with and
replaced with suprema and infima).
Then we have the following diagram:
What is at the node "other" in the diagram is unknown.
Conjecture "Other" is
.
![$ \lambda f\in\mathsf{FCD}: \top $](/files/tex/4a511edece8921fab6426695d3451efc024273a5.png)
Question What repeated applying of
and
to "other" leads to? Particularly, does repeated applying
and/or
to the node "other" lead to finite or infinite sets?
![$ \Phi_{\ast} $](/files/tex/c26f0d43856d263f335939666a99f483ffd09da8.png)
![$ \Phi^{\ast} $](/files/tex/cf6796f6de9023eb4c0ae3e69b8900a93e53fc6b.png)
![$ \Phi_{\ast} $](/files/tex/c26f0d43856d263f335939666a99f483ffd09da8.png)
![$ \Phi^{\ast} $](/files/tex/cf6796f6de9023eb4c0ae3e69b8900a93e53fc6b.png)
See Algebraic General Topology for definitions of used concepts.
The known part of the diagram is considered in this file.
Bibliography
* indicates original appearance(s) of problem.