
2 Open problems
Conjecture 1 Let ℧ be a set , F be the set of filters on ℧ ordered reverse to
set-theoretic inclusion, P be the set of principal filters on ℧, let n be an index
set. Consider the filtrator (F
n
; P
n
). If f is a multifuncoid of the form P
n
, then
E
∗
f is a multifuncoid of the form F
n
.
A similar conjecture about completary multifuncoids:
Conjecture 2 Let ℧ be a set , F be the set of filters on ℧ ordered reverse to
set-theoretic inclusion, P be the set of principal filters on ℧, let n be an index
set. Consider the filtrator (F
n
; P
n
). If f is a completary multifuncoid of the
form P
n
, then E
∗
f is a completary mult ifuncoid of the form F
n
.
A weaker conjecture:
Conjecture 3 Let ℧ be a set , F be the set of filters on ℧ ordered reverse to
set-theoretic inclusion, P be the set of principal filters on ℧, let n be an index
set. Consider the filtrator (F
n
; P
n
). If f is a completary multifuncoid of the
form P
n
, then E
∗
f is a multifuncoid of the form F
n
.
For finite n all three conjectures are equivalent.
For n = 0 the conjecture is trivial. For n = 1 it can be proved using the
theory of filters [2]. For n = 2 we can prove it using the theory of funcoids [1].
For card n > 3 (finite and infinite) the problem is open.
The full proo fs for c ard n 6 2 are presented below.
If a conjecture will be proved true, we may generalize it for a wider set of
filtrators.
3 Preliminary Results
3.1 Isomorphic filt rators
We will use the concept of isomorphic filtrators in the below proo fs.
An isom orphism from a filtrator (A
0
; Z
0
) to a filtrator (A
1
; Z
1
) is an order
embedding ϕ from A
0
to A
1
such that the image of Z
0
under ϕ is exactly Z
1
.
Two filtrators are isomorphic when there exist an isomorphism fr om one to
the other.
It is triv ial that neither the prope rty of be ing a multifuncoid, nor the result
of upgrading does change under an isomorphism.
3.2 The proof of the conjecture for card n 6 2
The below constitutes a proof of my conjecture for n ∈ {0, 1, 2} as well as n
being any set of cardinality card n 6 2, because a particular index se t doe s not
matter, just it’s cardinality.
Let Z = P
n
and A = F
n
.
4