I have just proven the following two new theorems:

Theorem Composition of complete reloids is complete.

Theorem $(\mathsf{RLD})_{\mathrm{out}} g \circ (\mathsf{RLD})_{\mathrm{out}} f = (\mathsf{RLD})_{\mathrm{out}} (g \circ f)$ if $f$ and $g$ are both complete funcoids (or both co-complete).

See this note for the proofs.

I’ve proved this conjecture (not a long standing conjecture, it took just one day to solve it) and found a stronger theorem than these propositions.

So my new theorem:

Theorem $(\mathsf{FCD})$ and $(\mathsf{RLD})_{\mathrm{out}}$ form mutually inverse bijections between complete reloids and complete funcoids.

For a proof see this note.

In this recent blog post I have formulated the conjecture:

Conjecture A funcoid $f$ is complete iff $f=(\mathsf{FCD}) g$ for a complete reloid $g$.

This conjecture has not been living long, I have quickly proved it in this note.

About new theorems in in this my blog post:

I’ve simplified this theorem:

Theorem A reloid $f$ is complete iff
$f = \bigsqcap^{\mathsf{RLD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, | \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\}$.

into

Theorem A reloid $f$ is complete iff $f=(\mathsf{RLD})_{\mathrm{out}} g$ for a complete funcoid $g$.

For a proof see this note.

The next theorem:

Theorem A funcoid $f$ is complete iff
$f = \bigsqcap^{\mathsf{FCD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, | \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\}$.

collapses into

Theorem $f=\bigsqcap^{\mathsf{FCD}} \mathrm{up}\, f$, what I proved long time ago.

So, I have removed this theorem from my writings.

Conjecture A funcoid $f$ is complete iff $f=(\mathsf{FCD}) g$ for a complete reloid $g$.

I’ve proved some new theorems. The proofs are currently available in this PDF file.

Theorem The set of funcoids is with separable core.

Theorem The set of funcoids is with co-separable core.

Theorem A funcoid $f$ is complete iff
$f = \bigsqcap^{\mathsf{FCD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, | \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\}$.

Theorem A reloid $f$ is complete iff
$f = \bigsqcap^{\mathsf{RLD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, | \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\}$.

It seems (I have not yet checked) that the following conjecture follows from the last theorem:

Conjecture Composition of complete reloids is complete.

I have published online a short article saying that the set of funcoids is isomorphic to the set of filters on a certain lattice.

But now I’ve realized that the counter-example is wrong.

So we can celebrate my new theorem. Now there is no reason to assume that it is false.

I will add more details to the article and keep you informed about it.

Earlier I have conjectured that the set of funcoids is order-isomorphic to the set of filters on the set of finite joins of funcoidal products of two principal filters. For an equivalent open problem I found a counterexample.

Now I propose another similar but weaker open problem:

Conjecture Let $U$ be a set. The set of funcoids on $U$ is order-isomorphic to the set of filters on the set $\Gamma$ (moreover the isomorphism is (possibly infinite) meet of the filter), where $\Gamma$ is the set of unions $\bigcup_{X\in S}(X\times Y_X)$ where $S$ is a finite partition of $U$ and $Y\in \mathscr{P} U$ for every $X\in S$

The last conjecture is equivalent to this question formulated in elementary terms. If you solve this (elementary) problem, it could be a major advance in mathematics.