# An important conjecture about funcoids. Version 2

This conjecture appeared to be false.

Now I propose an alternative conjecture:

Let $A$, $B$ be sets.

Conjecture Funcoids $f$ from $A$ to $B$ bijectively corresponds to the sets $R$ of pairs
$(\mathcal{X}; \mathcal{Y})$ of filters (on $A$ and $B$ correspondingly) that

1. $R$ is nonempty.
2. $R$ is a lower set.
3. every $\left\{ \mathcal{X} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$ is a dcpo for every $\mathcal{Y} \in \mathfrak{F}B$ and every $\left\{ \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$ is a dcpo for every $\mathcal{X} \in \mathfrak{F}A$

by the mutually inverse formulas:
$(\mathcal{X} ; \mathcal{Y}) \in R \Leftrightarrow \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \sqsubseteq f \quad \mathrm{and} \quad f = \bigsqcup^{\mathsf{FCD}} \left\{ \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$.