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 of funcoids between some sets and a set of principal funcoids (=binary relations)):

- .
- (for every natural ).
- There exists a funcoid such that .

and are obvious.

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. ;

5. (for every natural ).

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.