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.