Characterize the set $\{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_{\mathrm{out}} f\}$. (This seems a difficult problem.)

I have proved $(\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}} = \Omega^{\mathsf{RLD}}$ (where $\Omega^{\mathsf{FCD}}$ is a cofinite funcoid and $\Omega^{\mathsf{RLD}}$ is a cofinite reloid that is reloid defined by a cofinite filter).

The proof is currently available in this draft.

Note that in the previous draft there was a wrong formula for $(\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}}$.

I’ve found a typo in my math book.

I confused existential quantifiers with universal quantifiers in the section “Second product. Oblique product” in the chapter “Counter-examples about funcoids and reloids”.

I added more properties of cofinite funcoids to this draft.

I have described generalized cofinite filters (including the “cofinite funcoid”).

See the draft at http://www.mathematics21.org/binaries/addons.pdf

Define for posets with order $\sqsubseteq$:

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

Note that the above is a generalization of monotone Galois connections (with $\max$ and $\min$ 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 $\Phi_{\ast}$ and/or $\Phi^{\ast}$ 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 $f$ be an indexed family of funcoids. Let $\mathcal{F}$ be a filter on $\mathrm{dom}\, f$. $a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b \Leftrightarrow \exists N \in \mathcal{F} \forall i \in N : \mathrm{Pr}^{\mathsf{RLD}}_i\, a \mathrel{[f_i]} \mathrm{Pr}^{\mathsf{RLD}}_i\, b.$
for atomic reloids $a$ and $b$.

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 $\mathcal{F}$ 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.