Just a few minutes ago I’ve formulated a new important conjecture about funcoids:

Let , be sets.

**Conjecture** Funcoids from to bijectively corresponds to the sets of pairs

of filters (on and correspondingly) that

- is nonempty.
- is a lower set.
- (ordered pointwise) is a dcpo

by the mutually inverse formulas:

.

Oh, a trivial counter-example: . I will try to figure another axioms.

Oh, sorry, the counter-example is wrong. However it inspired me with another similar conjecture.

No, it is wrong. For a counter-example take .