Co-separability of filter objects (Solved)

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Unsorted
Keywords: filters
Recomm. for undergrads: no
Posted by: porton
on: November 29th, 2009
Solved by: 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.



* indicates original appearance(s) of problem.

A counterexample?

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

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

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

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?

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.

Oh, my mistake

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).

Comment viewing options

Select your preferred way to display the comments and click "Save settings" to activate your changes.