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 $latex S \in \mathscr{P} \mathfrak{A}$
$latex \mathrm{Cor}’ \bigsqcap^{\mathfrak{A}} S = \bigsqcap^{\mathfrak{Z}} \langle \mathrm{Cor}’ \rangle S.$
Particularly if $latex S \in \mathscr{P} \mathfrak{Z}$, this theorem reduced to the formula:
$latex \mathrm{Cor}’ \bigsqcap^{\mathfrak{A}} S = \bigsqcap^{\mathfrak{Z}} S.$
Particularly the above formulas hold for filters on complete lattices.
I’ve added these theorems and their simple consequences to my book.