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Difference between revisions of "Funcoid bases"

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Please write your research ideas at this wiki and as comments and trackbacks to [https://portonmath.wordpress.com/2017/04/11/research-in-the-middle-project/ this blog post].
 
Please write your research ideas at this wiki and as comments and trackbacks to [https://portonmath.wordpress.com/2017/04/11/research-in-the-middle-project/ this blog post].
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[[Funcoid bases/Basic results]]
  
 
Proposed ways to attack this conjecture:
 
Proposed ways to attack this conjecture:

Revision as of 20:21, 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 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. $\forall X_0,\dots,X_n \in S : \operatorname{up} (X \sqcap^{\mathsf{FCD}} \dots \sqcap^{\mathsf{FCD}} X_n) \subseteq S$ (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.

Funcoid bases/Basic results

Proposed ways to attack this conjecture:

$1\Rightarrow 2$:

$2\Rightarrow 3$: