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 multifuncoid of the form , then is a 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.

I found a really trivial proof of this conjecture. See this my draft article.

**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**A free star on a join-semilattice with least element 0 is a set such that and

**Definition**Let be a family of posets, ( has the order of function space of posets), , . Then

**Definition**Let is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form is an such that we have that:

- \item is a free star for every , .

\item is an upper set.

is a function space over a poset that is for .

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.