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 $latex \mathsf{FCD}$ of funcoids between some sets and a set $latex S$ of principal funcoids (=binary relations)):

- $latex \forall X, Y \in S : \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y) \subseteq S$.
- $latex \forall X_0,\dots,X_n \in S : \mathrm{up} (X \sqcap^{\mathsf{FCD}} \dots \sqcap^{\mathsf{FCD}} X_n) \subseteq S$ (for every natural $latex n$).
- There exists a funcoid $latex f\in\mathsf{FCD}$ such that $latex S=\mathrm{up}\, f$.

$latex 3\Rightarrow 2$ and $latex 2\Rightarrow 1$ 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 $latex \langle f\rangle$ with better known behavior of filters $latex \langle f\rangle x$ for an arbitrary ultrafilter $latex x$ (I remind that knowing $latex \langle f\rangle x$ for all ultrafilters $latex x$ on the domain, it’s possible to restore funcoid $latex f$) and then to replace $latex \langle X_0 \sqcap^{\mathsf{FCD}} \ldots \sqcap^{\mathsf{FCD}} X_n\rangle x$ with $latex \langle X_0 \rangle x \sqcap \dots \sqcap \langle X_n \rangle x$.

At https://conference.portonvictor.org/wiki/Funcoid_bases/Another_reduce_to_ultrafilters I introduce a proof attempt of the statement:

If $latex \forall X_0, \ldots, X_n \in S : \mathrm{up} (X_0 \sqcap^{\mathsf{FCD}} \ldots \sqcap^{\mathsf{FCD}} X_n) \subseteq S$ (for every natural $latex n$), then there exists a funcoid $latex f$ such that $latex S = \mathrm{up}\, f$.

I’ve published some easy basic results related to the conjecture at https://conference.portonvictor.org/wiki/Funcoid_bases/Basic_results

First I define $latex S’=S\cap\Gamma$. Second, I prove $latex \bigwedge^{\mathsf{FCD}} S’ = \bigwedge^{\mathsf{FCD}} S$.

At https://conference.portonvictor.org/wiki/Funcoid_bases/Proving_existence_of_funcoid_through_lattice_Gamma I tried to prove that $latex S$ is a an up of a funcoid (under another conjecture conditions). My attempted proof uses the lattice $latex \Gamma$ 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. $latex \forall X,Y\in S’: \mathrm{up}(X\sqcap Y)\subseteq S’$;

5. $latex \forall X_0,\dots X_n\in S’: \mathrm{up}(X_0\sqcap\dots\sqcap X_n)\subseteq S’$ (for every natural $latex n$).

The two above conditions 4 and 5 are each equivalent to $latex S’$ being a filter on the boolean lattice $latex \Gamma$.

It is easy to show that $latex S’$ being a filter is not enough for the (other) conditions of the conjecture to hold (for a counter-example consider $latex S\subseteq\Gamma$ and thus $latex S=S’$).

Probably the following is equivalent to the conditions of the conjecture: $latex S’$ is a filter on $latex \Gamma$ and $latex S$ is an upper set.

Added condition “4” defined above to the main wiki page. It is quite obvious that $latex 1\Rightarrow 4$ and $latex 3\Rightarrow 4$.

Should we also add to “4” the requirement for $latex S$ to be filter-closed? (see my book for a definition of being filter-closed).

The condition “$latex S’$ is a filter on the lattice $latex \Gamma$ and $latex S$ is an upper set” is not enough for existence of $latex f$ such that $latex S=\mathrm{up}\, f$. 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.