Conjecture The poset of multifuncoids of the form $ (\mathscr{P}\mho)^n $ is for every sets $ \mho $ and $ n $:

- atomic;

- atomistic. ...

Conjecture Let $ F $ is a family of multifuncoids such that each $ F_i $ is of the form $ \lambda j \in N \left( i \right) : \mathfrak{F} \left( U_j \right) $ where $ N \left( i \right) $ is an index set for every $ i $ and $ U_j $ is a set for every $ j $. Let every $ F_i = E^{\ast} f_i $ for some multifuncoid $ f_i $ of the form $ \lambda j \in N \left( i \right) : \mathfrak{P} \left( U_j \right) $ regarding the filtrator $ \left( \prod_{j \in N \left( i \right)} \mathfrak{F} \left( U_j \right) ; \prod_{j \in N \left( i \right)} \mathfrak{P} \left( U_j \right) \right) $. Let $ H $ is a graph-composition of $ F $ (regarding some partition $ G $ and external set $ Z $). Then there exist a multifuncoid $ h $ of the form $ \lambda j \in Z : \mathfrak{P} \left( U_j \right) $ such that $ H = E^{\ast} h $ regarding the filtrator $ \left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right) $. ...

Conjecture Let $ f_1 $ and $ f_2 $ are monovalued, entirely defined funcoids with $ \operatorname{Src}f_1=\operatorname{Src}f_2=A $. Then there exists a pointfree funcoid $ f_1 \times^{\left( D \right)} f_2 $ such that (for every filter $ x $ on $ A $) $$\left\langle f_1 \times^{\left( D \right)} f_2 \right\rangle x = \bigcup \left\{ \langle f_1\rangle X \times^{\mathsf{FCD}} \langle f_2\rangle X \; | \; X \in \mathrm{atoms}^{\mathfrak{A}} x \right\}.$$ (The join operation is taken on the lattice of filters with reversed order.) ...

Conjecture For composable reloids $ f $ and $ g $ it holds

- $ \operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f $ if $ f $ is a co-complete reloid;

- $ \operatorname{CoCompl} ( f \circ g) = f \circ \operatorname{CoCompl} g $ if $ f $ is a complete reloid; ...

Conjecture Every metamonovalued funcoid is monovalued. ...

Conjecture Every metamonovalued reloid is monovalued. ...

Let $ \delta $ be a proximity.

A set $ A $ is connected regarding $ \delta $ iff $ \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right) $.

Conjecture The following statements are equivalent for every endofuncoid $ \mu $ and a set $ U $:

- $ U $ is connected regarding $ \mu $. ...