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Difference between revisions of "Funcoid bases"

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*[[Funcoid bases/Reduce to meets of applying funcoids to ultrafilters]]
 
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$2\Rightarrow 3$:
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*[[Funcoid bases/Another reduce to ultrafilters]]
  
 
[[Category:Algebraic general topology]]
 
[[Category:Algebraic general topology]]
 
[[Category:Research in the middle]]
 
[[Category:Research in the middle]]

Revision as of 08:49, 13 April 2017

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

$3\Rightarrow 2$ and $2\Rightarrow 1$ are obvious.

Please write your research ideas at this wiki and as comments and trackbacks to this blog post.

Proposed ways to attack this conjecture:

$1\Rightarrow 2$:

$2\Rightarrow 3$: