Boolean lattices of multiply $\Omega$-foliated formations

In the context of a new functional approach to the study of classes of groups, V. A. Vedernikov and M. M. Sorokina introduced $\Omega$-foliated formations, which gave a possibility to systematise a wide class of formations of finite groups. In this paper, we study $n$-multiply $\Omega$-foliated formations with $r$-direction $\varphi$ such that $\varphi_0\leq\varphi$, $\varphi (A)\subseteq\mathfrak G_{A'}\mathfrak G_{A}$ for all $A\in\mathfrak I$ whose lattice of all $n$-multiply $\Omega$-foliated subformations with direction $\varphi$ is Boolean.