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Funcoid bases

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This page presents Victor Porton's conjectures about funcoid bases and related stuff as first defined in this draft document.

Please read Algebraic general topology book before attempt to participate in this research.

The main conjecture about funcoid bases is the following:

Conjecture The following are equivalent (for every lattice $\mathsf{FCD}$ of funcoids between some sets and a set $S$ of principal funcoids (=binary relations)):

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

$2\Rightarrow 3$ was disproved in Funcoid_bases/Disproof.

It yet remains the question whether the conditions "1" and "2" are equivalent.

The below are historical materials.

(Condition "4" was shown not to be equivalent to the above three conditions.)

$3\Rightarrow 2$ and $2\Rightarrow 1$ are obvious. It's also easy to prove that $1\Rightarrow 4$ (taking into account that $\Gamma$ is a sublattice of $\mathsf{FCD}$).

$3\Rightarrow 4$ because $X,Y\in S\cap\Gamma \Rightarrow X,Y\in\Gamma\land X,Y\sqsupseteq f \Rightarrow X\sqcap Y\in\Gamma\land X\sqcap Y\sqsupseteq f \Rightarrow X\sqcap Y\in S\cap\Gamma$.

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Funcoid bases/Basic results

Proposed ways to attack this conjecture:

$1\Rightarrow 2$:

$2\Rightarrow 3$: