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




Re: On an open problem about filters
Posted:
Sep 19, 2013 11:33 PM


On Thursday, September 19, 2013 9:44:35 PM UTC5, 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 {GoFF 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 > > > settheoretic inclusion.) > > > > > > Can anybody help me to solve this important problem? > > > > Yes, unreverse the order. > > > > > First reading (a part of) my book may help you to solve this problem: > > > http://www.mathematics21.org/binaries/volume1.pdf > > > > Didn't you unreverse the reversed order for your book?
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