Principal filters are center – solved
I have proved this conjecture:
Theorem 1 If is the set of filter objects on a set then is the center of the lattice . (Or equivalently: The set of principal filters on a set is the center of the lattice of all filters on .)
Proof: I will denote the center of the lattice . I will denote the set of atoms of a lattice under its element .
Let . Then exists such that and . Consequently, there are such that ; we have also . Suppose . Then (because for is true the disjunct propery of Wallman, see ) exists such that . We can conclude also . Thus and consequently what is a contradiction. We have .
Let now . Then and . Thus ; (used formulas from ). We have shown that .
This theorem may be generalized for a wider class of filters on lattices than only filters on lattices of a subsets of some set.