# 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 [1]) exists such that . We can conclude also . Thus and consequently what is a contradiction. We have .

Let now . Then and . Thus ; (used formulas from [1]). 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.

[1] Victor Porton. Funcoids and Reloids. http://www.mathematics21.org/binaries/set-filters.pdf

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