# Upgrading a completary multifuncoid

Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .

**Conjecture**If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.

**Definition**A filtrator is a pair of a poset and its subset .

Having fixed a filtrator, we define:

**Definition**for every .

**Definition**(upgrading the set ) for every .

**Definition**Let is a family of join-semilattice. A completary multifuncoid of the form is an such that we have that:

- \item for every .

\item If and for some then .

is a function space over a poset that is for .

For finite this problem is equivalent to Upgrading a multifuncoid .

It is not hard to prove this conjecture for the case using the techniques from this my article. But I failed to prove it for and above.

## Bibliography

* indicates original appearance(s) of problem.