I start the “research-in-the middle” project (an outlaw offspring of Polymath Project) introducing to your attention the following conjecture:
Conjecture The following are equivalent (for every lattice 
.
(for every natural
).
- There exists a funcoid
such that
.
I welcome you to actively participate in the research!
Please write your comments and idea both in the wiki and as comments and trackbacks to this blog post.
I present an attempted proof in the wiki.
The idea behind this attempted proof is to reduce behavior of funcoids
with better known behavior of filters 
for an arbitrary ultrafilter 
(I remind that knowing 
for all ultrafilters 
on the domain, it’s possible to restore funcoid 
) and then to replace 
with 
.
At https://conference.portonvictor.org/wiki/Funcoid_bases/Another_reduce_to_ultrafilters I introduce a proof attempt of the statement:
If
(for every natural 
), then there exists a funcoid 
such that 
.
I’ve published some easy basic results related to the conjecture at https://conference.portonvictor.org/wiki/Funcoid_bases/Basic_results
First I define
. Second, I prove 
.
At https://conference.portonvictor.org/wiki/Funcoid_bases/Proving_existence_of_funcoid_through_lattice_Gamma I tried to prove that
is a an up of a funcoid (under another conjecture conditions). My attempted proof uses the lattice 
from the chapter “Funcoids are filters” of my book
I propose also the following two conditions (possibly) equivalent to the conditions mentioned in the original conjecture:
4.
;
(for every natural 
).
5.
The two above conditions 4 and 5 are each equivalent to
being a filter on the boolean lattice 
.
It is easy to show that
being a filter is not enough for the (other) conditions of the conjecture to hold (for a counter-example consider 
and thus 
).
Probably the following is equivalent to the conditions of the conjecture:
is a filter on 
and 
is an upper set.
Added condition “4” defined above to the main wiki page. It is quite obvious that
and 
.
Should we also add to “4” the requirement for
to be filter-closed? (see my book for a definition of being filter-closed).
The condition “
is a filter on the lattice 
and 
is an upper set” is not enough for existence of 
such that 
. See https://conference.portonvictor.org/wiki/Funcoid_bases/Failed_condition in the wiki. So the condition “4” is removed from consideration.
Can the same counter-example as in https://conference.portonvictor.org/wiki/Funcoid_bases/Failed_condition (the topic of the previous comment) be applied to some implications between conditions 1, 2, 3?
The conjecture was declined with a counter-example https://conference.portonvictor.org/wiki/Funcoid_bases/Disproof
It yet remains the question whether the condition “1” implies “2”.
The proof at https://conference.portonvictor.org/wiki/Funcoid_bases/Disproof was with an error, but the proof idea was right. Now it contains the corrected proof.