For not empty A subset S, let F_A be the principal filter for S generated by {A}. For x in S, let F_x = F_{x} be the principal ultrafilter for S generated by {{x}}.

Theorem. If F is a filter, then F = /\{ G ultrafilter | F subset G } Proposition. If A not empty, then F_A = /\{ F_x | x in A }.

Question. If A not empty and F a filter with F_A subset F, does F = /\{ F_x | x in A, F subset F_x }?