Today I have noticed a new simple to prove theorem which is missing in my article:

**Theorem** If $latex ( \mathfrak{A}; \mathfrak{Z})$ is a join-closed filtrator and both $latex \mathfrak{A}$ and $latex \mathfrak{Z}$ are complete lattices, then for every $latex S \in \mathscr{P} \mathfrak{A}$

Particularly if $latex S \in \mathscr{P} \mathfrak{Z}$, this theorem reduced to the formula:

Particularly the above formulas hold for filters on complete lattices.

I’ve added these theorems and their simple consequences to my book.