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Note that as it is easy to prove for every set of ultrafilters.

Does this funcoid posses interesting properties? Can it be used to prove any open problem?

What is its behavior on non-principal filters?

I started researching properties of this weird funcoid in this PDF file.

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See the definition and the “curious” proposition in this draft.

Note that I work on another projects and may be not very active in researching funcoidal groups in near time.

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**Theorem** Let be a distributive lattice with least element. Let . If exists, then also exists and .

The user quasi of Math.SE has helped me with the proof.

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The proof was with an error (see the previous edited post)!

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I knew that composition of two complete funcoids is complete. But now I’ve found that for to be complete it’s enough to be complete.

The proof which I missed for years is rather trivial:

Thus is complete.

I will amend my book when (sic!) it will be complete.

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**Proposition** For , a finite set and a function there exists (obviously unique) such that for and for .

This funcoid is determined by the formula

and its corollary:

**Corollary** If , , and , then there exists an (obviously unique) such that for all ultrafilters except of and .

This funcoid is determined by the formula

.

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The error was that I claimed that infimum of a greater set is greater (while in reality it’s lesser).

I will delete the erroneous theorem from my book soon.

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Alleged counter-example:

and for infinite sets and .

I am now attempting to locate the error in the proof.

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**Theorem** Let and . Then there is an (obviously unique) funcoid such that for nontrivial ultrafilters and for

After I started to prove it, it took about a hour or like this to finish the proof.

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