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)):

  1. $latex \forall X, Y \in S : \mathrm{up} (X \sqcap^{\mathsf{FCD}} Y) \subseteq S$.
  2. $latex \forall X_0,\dots,X_n \in S : \mathrm{up} (X \sqcap^{\mathsf{FCD}} \dots \sqcap^{\mathsf{FCD}} X_n) \subseteq S$ (for every natural $latex n$).
  3. 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.

13 thoughts on “A new research project (a conjecture about funcoids)

  1. 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 $latex \langle X_0 \rangle x \sqcap \dots \sqcap \langle X_n \rangle x$.

  2. 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 $latex n$).
  3. 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.

  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).

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