Characterize the set . (This seems a difficult problem.)
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 get the following diagram (see this PDF file for a proof):
It is yet unknown what will happen if we keep apply and/or to the node “other”. Will this lead to a finite or infinite set?
The following is one of a few (possibly non-equivalent) definitions of products of funcoids:
Definition Let be an indexed family of funcoids. Let be a filter on .
for atomic reloids and .
Today I have proved that this really defines a funcoid. Currently the proof is present in draft of the second volume of my book,
A probably especially interesting case is if is the cofinite filter. In this way we get something similar to Tychonoff product of topological spaces.
This may possibly have some use in study of compact funcoids.