**Theorem** Let be a set of binary relations. If for every we have then there exists a funcoid such that .

The proof (currently available in this PDF file) is based on “Funcoids are filters” chapter of my book.

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Counterexample: Take . We know that is not a filter base. But it is trivial to prove that is a base of the funcoid .

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It was easy to show:

**Proposition** A set of binary relations is a base of a funcoid iff it is a base of .

Today I’ve proved the following important theorem:

**Theorem** If is a filter base on the set of funcoids then is a base of .

The proof is currently located in this PDF file.

It is yet unknown whether the converse theorem holds, that is whether every base of a funcoid is a filter base on the set of funcoids.

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**Statement** or equivalently: If then there exists , such that .

But now I’ve noticed that the proof **was with an error!** So it is again a conjecture.

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**Lemma** Let for every and there is a such that .

Then for every and there is a such that .

I spent much time (probably a few hours) to prove it, but the found proof is really simple, almost trivial.

The proof is currently located in this PDF file.

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It seems easy to generalize it for more general lattices than the lattice of funcoids, what I hope to do a little later.

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**Conjecture** for all funcoids , (with corresponding sources and destinations).

Looks trivial? But how to (dis)prove it?

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This seems an interesting research by itself, but I started to develop it as a way to prove this conjecture.

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