On maximal subformations of $n$-multiple $\Omega$-foliated formations of finite groups

Only finite groups are considered in the article. Among the classes of groups the central place is occupied by classes closed regarding homomorphic images and subdirect products which are called formations. We study $\Omega$-foliateded formations constructed by V. A. Vedernikov in 1999 where $\Omega$ is a nonempty subclass of the class $\frak I$ of all simple groups. $\Omega$-Foliated formations are defined by two functions — an $\Omega$-satellite $f: \Omega \cup \{\Omega '\} \rightarrow \{$formations$ \}$ and a direction $\varphi: \frak I \rightarrow \{$nonempty Fitting formations$\}$. The conception of multiple locality introduced by A. N. Skiba in 1987 for formations and further developed for many other classes of groups, as applied to $\Omega$-foliated formations is as follows: every formation is considered to be $0$-multiple $\Omega$-foliated with a direction $\varphi$; an $\Omega$-foliated formation with a direction $\varphi$ is called an $n$-multiple $\Omega$-foliated formation where $n$ is a positive integer if it has such an $\Omega$-satellite all nonempty values of which are $(n-1)$-multiple $\Omega$-foliated formations with the direction $\varphi$. The aim of this work is to study the properties of maximal $n$-multiple $\Omega$-foliated subformations of a given $n$-multiple $\Omega$-foliated formation. We use classical methods of the theory of groups, of the theory of classes of groups, as well as methods of the general theory of lattices. In the paper we have established the existence of maximal $n$-multiple $\Omega$-foliated subformations for the formations with certain properties, we have obtained the characterization of the formation $\Phi_{_{n\Omega\varphi}} (\frak F)$ which is the intersection of all maximal $n$-multiple $\Omega$-foliated subformations of the formation $\frak F$, and we have revealed the relation between a maximal inner $\Omega$-satellite of $1$-multiple $\Omega$-foliated formation and a maximal inner $\Omega$-satellite of its maximal $1$-multiple $\Omega$-foliated subformation. The results will be useful in studying the inner structure of formations of finite groups, in particular, in studying the maximal chains of subformations and in establishing the lattice properties of formations.