Boolean lattices of $n$-multiply $\omega\sigma$-fibered fitting classes

Let $\mathbb{N}$ be the set of all natural numbers. Consider all definitions and results taking into account the partitioning of the area for determining satellites and directions. An arbitrary Fitting class is considered a $0$-multiply fibered Fitting class; for $n$ equal to or greater than $1$, a Fitting class is said to be $n$-multiply fibered if it has at least one satellite $f$, all non-empty values which are $(n-1)$-multiply fibered Fitting classes. The main result of this work is a description of $n$-multiply fibered Fitting classes, for which the lattice of all $n$-multiply fibered Fitting subclasses is Boolean. It is shown that such classes are representable in the form of a direct decomposition of lattice atoms. In this article, direct decompositions of $n$-multiply fibered Fitting classes are studied in detail. The direction of these classes is the main one, and is taken from the segment between the directions of the complete and local Fitting classes. Particular results for $n$-multiply complete and $n$-multiply local Fitting classes are obtained as corollaries of the corresponding theorems. When proving the statements, the methods of counter inclusions and mathematical induction were used. The results obtained can be used in the further study of Boolean lattices of $n$-multiply fibered Fitting classes with directions from other intervals, as well as Stone lattices of $n$-multiply fibered Fitting classes.