Co-separability of filter objects (Solved)

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Posted	by:	porton
	on:	November 29th, 2009

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Porton, Victor

Conjecture Let $a$ and $b$ are filters on a set $U$ and $a\cap b = \{U\}$ . Then $\exists A\in a,B\in b: (\forall X\in a: A\subseteq X \wedge \forall Y\in b: B\subseteq Y \wedge A \cup B = U).$

See here for some equivalent reformulations of this problem.

This problem (in fact, a little more general version of a problem equivalent to this problem) was solved by the problem author. See here for the solution.

Maybe this problem should be moved to "second-tier" because its solution is simple.

Bibliography

*Victor Porton. Open problem: co-separability of filter objects

* indicates original appearance(s) of problem.

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A counterexample?

On September 7th, 2010 Robert Samal says:

From the link it seems you have proved the result. What about the following what seems to be a counterexample?

$U = \mathbb N$ (the set of integers), $a = \{ U \}$ , $b =$ the set of all infinite sets of integers

Now there is no set $B$ that would be minimal in $b$ ...

Your example is wrong

On September 7th, 2010 porton says:

The set of all infinite sets of integers is not a filter. For example $\{0,2,4,\dots\} \cap \{1,3,5,\dots\} = \emptyset$ .

I haven't read your comment further.

Correction

On September 8th, 2010 Robert Samal says:

Sorry, I was too hasty. What I meant is that $b$ is a "nontrivial ultrafilter" (wikipedia page ultrafilter) calls this "non-principal ultrafilter".

No counterexamples, it is proved

On September 8th, 2010 porton says:

Then take $A=U$ and $B=\emptyset$ (I do not require filters to be proper).

Robert, why you are trying to find a counter-example for a proved theorem?

Here is the proof.

--
Victor Porton - http://www.mathematics21.org

Is it really?

On September 10th, 2010 Robert Samal says:

You require that $B \in b$ , and my filter $b$ does not contain empty set. I'm trying to find a counter-example because either I misunderstand the statement of the theorem, or the theorem is false.

Some proofs just happen to have mistakes. Unfortunately, I don't understand yours, it apparently uses lot of notation (up, down, Cor, ...) that I'm unfamiliar with.

I made a mistake in the statement of the conjecture as published on OPG. I corrected the problem statement both on OPG and on my blog. It should be $\exists A,B\in\mathscr{P}U$ rather than $\exists A\in a,B\in b$ .

Indeed the equivalent reformulations of the theorem are correct and my proof (of a more general statement than this theorem) is not affected by the above mentioned error.

Robert, you do not understand me because I introduced new notations (that up, down, Cor, etc.) You may wish to read my preprint about these things (filters on posets and generalizations).

Open Problem Garden

Co-separability of filter objects (Solved)

Bibliography

A counterexample?

Your example is wrong

Correction

No counterexamples, it is proved

Is it really?

Oh, my mistake

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