
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)?

