# Conjecture: Upgrading a multifuncoid

This short article is the first my public writing where I introduce the concept of *multidimensional funcoid* which I am investigating now.

But the main purpose of this article is to formulate a conjecture (see below). This is the shortest possible writing enough to explain my conjecture to every mathematician.

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

If you solve this my open problem, please send me the solution.

**Definition 1** * A filtrator is a pair of a poset and its subset . *

Having fixed a filtrator, we define:

**Definition 2** * for every . *

**Definition 3** * ( upgrading the set ) for every . *

**Definition 4** * A free star on a join-semilattice with least element 0 is a set such that and *

* *

**Definition 5** * Let be a family of posets, ( has the order of function space of posets), , . Then *

* *

**Definition 6** * Let is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form is an such that we have that:*

- is a free star for every , .
- is an upper set.

* *

is a function space over a poset that is for .

**Conjecture 7** * Let be a set, be the set of f.o. on , be the set of principal f.o. on , let be an index set. Consider the filtrator . If is a multifuncoid of the form , then is a multifuncoid of the form . *

It is not hard to prove this conjecture for the case using the techniques from this my article. But it’s not easy to prove it for and above. I failed to find a general solution.

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