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Topic: On an open problem about filters
Replies: 10   Last Post: Sep 23, 2013 7:21 AM

 Messages: [ Previous | Next ]
 donstockbauer@hotmail.com Posts: 1,412 Registered: 8/13/05
Re: On an open problem about filters
Posted: Sep 19, 2013 11:33 PM

On Thursday, September 19, 2013 9:44:35 PM UTC-5, William Elliot wrote:
> On Thu, 19 Sep 2013, Victor Porton wrote:
>
>
>

> > Let A is the complete lattice of all filters on a cartesian product UxU for
>
> > some set U. (I allow the improper filter and so it is really a complete
>
> > lattice.)
>
> >
>
> > Composition gof of filters f and g on UxU is defined as the filter having
>
> > the base {GoF|F in f, G in g} (where GoF denotes composition of binary
>
> > relations).
>
> >
>
> > Can you prove that for a principal filter f and a set T of filters (on AxA)
>
> > the following formula holds?
>
> >
>
> > f o /\T = /\ { fog | g in T }
>
>
>
> /\T is, of course, the great intersection of T.
>
>
>
> If R is a relation for XxY and C a collection of relations for YxZ, does
>
> . . R o /\C = /\{ RoS | S in C }?
>
>
>

> > This is an open problem introduced by me.
>
> >
>
> > Yesterday, I had something that seemed to me to be a brilliant idea how to
>
> > solve this conjecture. But I've failed to produce a complete proof.
>
> >
>
> > For my partial proof (with this "brilliant" idea) see:
>
> > http://www.mathematics21.org/binaries/decomposition.pdf
>
> >
>
> Only rarely do I mess with pdf.
>
>
>

> > (Note that in my writings the order of the lattice of filters is REVERSE to
>
> > set-theoretic inclusion.)
>
> >
>
> > Can anybody help me to solve this important problem?
>
>
>
> Yes, unreverse the order.
>
>
>

>
> > http://www.mathematics21.org/binaries/volume-1.pdf
>
>
>
> Didn't you unreverse the reversed order for your book?

Buy yourself a Fram one and change it as soon as possible before engine damage results.

Date Subject Author
9/19/13 Victor Porton
9/19/13 William Elliot
9/19/13 donstockbauer@hotmail.com
9/20/13 Victor Porton
9/20/13 William Elliot
9/20/13 Victor Porton
9/20/13 William Elliot
9/21/13 Victor Porton
9/22/13 William Elliot
9/23/13 Victor Porton
9/22/13 William Elliot