I erroneously concluded (section “Distributivity of the Lattice of Filters” of my book) that

the base of every primary filtrator over a distributive lattice which is an ideal base is a co-frame.

Really it can be not a complete lattice, as in the example of the lattice of the poset of all small (belonging to a Grothendieck universe) sets.

I will update my book soon.

I’ve corrected this error.