Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Topic: Ultrafilter theorem refined
Replies: 27   Last Post: Jan 4, 2014 9:42 PM

 Messages: [ Previous | Next ]
 William Elliot Posts: 1,238 Registered: 1/8/12
Ultrafilter theorem refined
Posted: Dec 28, 2013 9:43 PM

Let S be a set, F_A = { B subset S | A subset B }
the principal filter for S generated by {A}.
The principal ultrafilters are F_x = F_{x}, for x in S.

The usual ultrafilter theorem is for a filter F,
F = /\{ G ultrafilter | F subset G }

This leads to a corollary, if F is a free filter, then
F = /\{ G free ultrafilter | F subset G }.

Directly one can show that if F_A is a principal ultrafilter,
F_A = { F_x | x in A }.

There remains the case when F is not free and not principal.

From the ultratheorem come another corollary.
If A = /\F not empty, then
F = { F_x | x in A } /\ { G ultrafilter | F subset G }
. = F_A /\ G ultrafilter | F subset G }.

Can that be refined for a filter F for S, to:
F = F_/\F /\ { G free ultrafilter | F subset G }

with the understanding F_(empty set) = P(S)?

Date Subject Author
12/28/13 William Elliot
12/29/13 William Elliot
12/29/13 Victor Porton
12/29/13 Victor Porton
12/29/13 William Elliot
12/30/13 Victor Porton
12/30/13 William Elliot
1/1/14 Niels Diepeveen
1/2/14 William Elliot
1/2/14 Niels Diepeveen
1/2/14 William Elliot
1/3/14 Niels Diepeveen
1/2/14 Victor Porton
1/4/14 William Elliot
1/4/14 Victor Porton
1/4/14 William Elliot
1/2/14 Victor Porton
1/2/14 Niels Diepeveen
1/2/14 Victor Porton
1/2/14 Victor Porton
1/2/14 William Elliot
1/3/14 Victor Porton
1/3/14 Niels Diepeveen
1/4/14 Victor Porton
1/4/14 Victor Porton
1/4/14 Victor Porton
1/4/14 Niels Diepeveen
12/29/13 Victor Porton