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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